DESIGN-ORIENTED TRANSLATORS FOR AUTOMOTIVE JOINTS

DESIGN-ORIENTED TRANSLATORS FOR AUTOMOTIVE JOINTS by Luohui Long DISSERTATION SUBMITTED TO THE FACULTY OF VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN AEROSPACE ENGINEERING APPROVED E. Nikolaidis, Chairman O. F. Hughes E. R. Johnson R. Kapania C. E. Knight C. P. Agrawal September 1998 Blacksburg, Virginia DESIGN-ORIENTED TRANSLATORS FOR AUTOMOTIVE JOINTS by Luohui Long Committee Chairman: Efstratios Nikolaidis Aerospace Engineering (ABSTRACT) A hierarchical approach is typically followed in design of consumer products. First, a manufacturer sets performance targets for the whole system according to customer surveys and benchmarking of competitors’ products. Then, designers cascade these targets to the subsystems or the components using a very simplified model of the overall system. Then, they try to design the components so that they meet these targets. It is important to have efficient tools that check if a set of performance targets for a component corresponds to a feasible design and determine the dimensions and mass of this design. This dissertation presents a methodology for developing two tools that link performance targets for a design to design variables that specify the geometry of the design. The first tool (called translator A) predicts the stiffness and mass of an automotive joint, whose geometry is specified, almost instantaneously. The second tool (called translator B) finds the most efficient, feasible design whose performance characteristics are close to given performance targets. The development of the two translators involves the following steps. First, an automotive joint is parameterized. A set of physical parameters are identified that can ii completely describe the geometry of the joint. These parameters should be easily understood by designers. Then, a parametric model is created using a CAD program, such as Pro/Engineer or I-Deas. The parametric model can account for different types of construction, and includes relations for styling, packaging, and manufacturing constraints. A database is created for each joint using the results from finite element analysis of hundreds or thousands of joint designs. The elements of the database serve as examples for developing Translator A. Response surface polynomials and neural networks are used to develop translator A. Stepwise regression is used in this study to rank the design variables in terms of importance and to obtain the best regression model. Translator B uses optimization to find the most efficient design. It analyzes a large number of designs efficiently using Translator A. The modified feasible direction method and sequential linear programming are used in developing translator B. The objective of translator B is to minimize the mass of the joint and the difference of the stiffness from a given target while satisfying styling, manufacturing and packaging constraints. The methodologies for Translators A and B are applied to the B-pillar to rocker and A-pillar to roof rail joints. Translator B is demonstrated by redesigning two joints of actual cars. Translator B is validated by checking the performance and mass of the optimum designs using finite element analysis. This study also compares neural networks and response surface polynomials. It shows that they are almost equally accurate when they are used in both analysis and design of joints. iii Acknowledgements During the last four years I was fortunate to work with the research group of professors, and graduate students at the Aerospace and Ocean Engineering Department of Virginia Polytechnic Institute and State University, and engineers from the CAE department of Ford Motor Company. The unique environment of cooperation and support both in studies and in research encouraged and greatly contributed to my academic and professional growth. In the very first lines of this work I would like to express my deep gratitude and thankfulness to my advisor and committee chairman, Dr. Efstratios Nikolaidis. From my very first days at Virginia Tech he was a source of guidance and support to me. No matter what kind of problem was discussed, his advice and encouragement always provided new perspectives. The expertise that he shared with me remains a tremendous source for the professional growth to me. I am very thankful for everything he has done for me. I am most thankful to all the members of my committee: Dr. E. Johnson, Dr. O. Hughes, Dr. R. Kapania, Dr. C. Knight, Dr. C. Agrawal and Dr. Z. Grdal. Their technical expertise and assistance played the key role in the progress of my research. All their extraordinary help is greatly appreciated. The great contribution to everything I have done was made by my many friends both undergraduate and graduate students from Virginia Tech. It is impossible to list all of them over here, but particular thanks are addressed to Q. Ling, S. Murphy, N. Guyot, and M. Beeken. Financial support for this research project was provided by the Department of Aerospace and Ocean Engineering and Ford Motor Company. I am thankful to our contract monitors, Dr. C. Agrawal, Dr. S. Patill, Dr. K. Sohn, Mr. B. Eagan, Mr. P. Geck iv and S. Jasuja who provided technical and financial support. The author would also like to acknowledge the support of Dr. S. Kelkar of Ford Motor Company, who understood the potential of this project in its early stages and helped secure support. Foremost, I would like to thank my family who has been unfailingly supportive throughout my educational career. I dedicate this work to them. v Contents Abstract Acknowledgements Nomenclature 1 1.1 1.2 Introduction ii iv xxi 1 Definition and Significance of the Problem ........................................................ 1 Previous Work ..................................................................................................... 4 1.2.1 Structural Analysis of Joints in Automotive Structures ........................... 4 1.2.2 Response Surface Polynomials in Structural Analysis and Design ......... 5 1.2.3 Neural Networks in Structural Analysis and Design ................................ 7 1.2.4 Optimization of Automotive Structures .................................................. 13 1.2.5 CAD Software ........................................................................................ 15 1.2.6 Neural Networks or Polynomials versus Finite Element Analysis in Optimization ........................................................................................... 16 1.3 1.4 2 Objectives and Benefits of the Project Presented in this Dissertation ............... 18 Outline of the Dissertation ................................................................................ 21 30 Methodology for Developing Translator A 2.1 2.2 2.3 2.4 Introduction ....................................................................................................... 30 Parametric Models ........................................................................................... 31 Database for Developing Translator A ........................................................... 33 Developing and Validating a Response Surface Polynomial ........................... 36 2.4.1 2.4.2 2.4.3 2.4.4 Determining the Most Important Design Variables .............................. 36 Choosing a Polynomial Regression Model ........................................ 37 Criteria for Selecting Polynomial Translators ..................................... 40 Validation of Polynomials ................................................................... 44 vi 2.5 Developing and Validating a Neural Network ................................................ 45 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 Transfer Function ................................................................................ 46 Choosing Inputs and Outputs of a Neural Network ............................ 47 Normalization of the Input-Output Sets .............................................. 48 Determining the Number of Neurons in the Hidden Layer .................. 49 Method of Determining When to Stop Training .................................. 50 Method of Testing the Generalization Performance of the Trained Neural Network .................................................................................. 52 3 Methodology For Developing Translator B 3.1 3.2 59 Introduction ..................................................................................................... 59 Formulation of the Optimization Problem ....................................................... 60 3.2.1 3.2.2 3.2.3 Selection of Design Variables .............................................................. 61 Objective Function ............................................................................... 62 Constraints ............................................................................................. 64 Modified Feasible Direction Method ................................................... 66 Sequential Linear Programming ........................................................... 67 3.3 Optimization Algorithms used in Translator B ................................................ 65 3.3.1 3.3.2 3.4 4 Validation of Results of Translator B .............................................................. 68 71 Developing Translator A for the B-Pillar to Rocker Joint 4.1 4.2 Introduction ...................................................................................................... 71 Description of the B-pillar to rocker Joint ....................................................... 71 4.2.1 Types of Reinforcements ..................................................................... 73 4.2.2 4.2.3 Types of Rocker Cross Section ............................................................ 73 Type of Construction used in Developing Translator A ...................... 74 4.3 4.4 Developing a Database for Translator A............................................................ 74 Developing Translator A .................................................................................. 76 4.4.1 4.4.2 4.4.3 Ranking Important Design Variables ................................................... 76 Developing Polynomial Translators ..................................................... 77 Developing Neural Networks ............................................................... 78 4.5 Results and Discussion .................................................................................... 80 vii 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 5 Comparison of Predictions from Translators A and FEA Results ......... 80 I/O Stiffness............................................................................................ 81 F/A Stiffness ......................................................................................... 81 Torsion Stiffness .................................................................................. 82 Mass ..................................................................................................... 83 Validation ............................................................................................. 83 Conclusions .......................................................................................... 85 127 Developing Translator A for the A-Pillar to Roof Rail Joint 5.1 5.2 Introduction ................................................................................................... 127 Description of the A-pillar to Roof Rail Joint .............................................. 127 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7 5.2.8 Design Variables Defining the Orientation and the Position of the Branches .............................................................................................. 129 Dimensions Defining the Cross Section of the Header ....................... 130 Dimensions Defining the Cross Section of the Roof Rail.................... 131 Dimensions Defining the Cross Section of the A-pillar ...................... 131 Blending Radii...................................................................................... 131 Connections.......................................................................................... 131 Other Dimensions................................................................................. 132 Parametric Model and FEA of the A-pillar to Roof Rail Joint ............ 132 5.3 5.4 Developing a Database .................................................................................. 134 Developing Translator A ................................................................................ 135 5.4.1 5.4.2 5.4.3 Ranking Important Design Variables ................................................. 135 Polynomial Translators ...................................................................... 136 Neural Network Translators ............................................................... 140 I/O Stiffness ........................................................................................ 141 F/A Stiffness ....................................................................................... 142 Torsion Stiffness ................................................................................ 142 Mass ................................................................................................... 143 Validation ........................................................................................... 143 Conclusions ........................................................................................ 146 5.5 Results and Discussion ................................................................................. 141 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 viii 6 Developing Translator B for an Actual B-Pillar to Rocker Joint 6.1 6.2 Formulation of the Optimization Problem for Developing a Translator B 195 Introduction ..................................................................................................... 195 for a B-pillar to Rocker Joint .......................................................................... 196 6.2.1 6.2.2 6.2.3 6.2.4 Definition of the Problem .................................................................... 196 Design Variables ................................................................................ 197 Objective Function ............................................................................. 198 Constraints .......................................................................................... 199 Checking the Convergence of the Optimization Program ................. 209 Comparison of Results of Translator B with FEA Results ................ 211 Redesign of the Joints of two Cars Using Translator B ....................... 213 Parametric Study ................................................................................ 215 Discussion of Results ......................................................................... 216 261 6.3 Results and Discussion ................................................................................... 209 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 7 Developing Translator B for an Actual A-Pillar to Roof Rail Joint 7.1 7.2 Formulation of the Optimization Problem for Developing a Translator B Introduction ................................................................................................... 261 for an A-pillar to Roof Rail Joint .................................................................... 262 7.2.1 7.2.2 7.2.3 7.2.4 Definition of the Problem .................................................................... 262 Design Variables ................................................................................ 263 Objective Function ............................................................................. 265 Constraints .......................................................................................... 265 Checking the Convergence of the Optimization Program ................ 277 Comparison of the Results of Translator B with FEA Results ............ 278 Redesign of the Joints of two Cars Using Translator B ..................... 281 Parametric Study ............................................................................... 282 Discussion of Results ......................................................................... 284 7.3 Results and Discussion ................................................................................... 277 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 ix 8 Conclusions 8.1 8.2 8.3 347 Conclusions .................................................................................................... 347 Recommendation for Future Work ................................................................ 350 Deliverables...................................................................................................... 352 353 361 Bibliography Vita x List of Tables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 Comparison of Stiffness and Mass for Different Types of Rocker Cross Section................................................................................................................... 86 Measured Dimensions for B-pillar to Rocker Joint ............................................. 86 Ranges of Design Variables Used in Creating the Database ............................... 88 Ranking of Important Dimensions for I/O Stiffness ............................................ 90 Ranking of Important Dimensions for F/A Stiffness ........................................... 91 Ranking of Important Dimensions for Torsion Stiffness ...................................... 91 Ranking of Important Dimensions for Mass ........................................................ 92 Comparison of Different Models for B-pillar to Rocker Joint ............................. 92 Effects of Mesh Size on FEA Results ................................................................ 147 Comparison of FEA Results and Experimental Results ..................................... 147 Measured Dimensions for A-pillar to Roof Rail Joints ..................................... 148 Ranges of Design Variables for A-pillar to Roof Rail Joint .............................. 150 Ranking of Important Design Variables for the I/O Stiffness of A-pillar to Roof Rail Joint ................................................................................................... 152 Ranking of Important Design Variables for the F/A Stiffness of A-pillar to Roof Rail ............................................................................................................ 153 Ranking of Important Design Variables for the Torsion Stiffness of A-pillar to Roof Rail Joint ................................................................................................ 154 Ranking of Important Design Variables for the Mass of A-pillar to Roof Rail Joint ............................................................................................................. 155 Comparison of Results from Different Methods for the I/O Stiffness of Apillar to Roof Rail Joint ...................................................................................... 156 Comparison of Results from Different Methods for the F/A Stiffness of Apillar to Roof Rail Joint ...................................................................................... 156 Comparison of Results from Different Methods for the Torsion Stiffness of A-pillar to Roof Rail Joint .................................................................................. 156 xi 5.12 6.1 6.2 6.3 6.4a 6.4b 6.4c 6.5a 6.5b 6.5c 6.5d 7.1 7.2 7.3a 7.3b 7.3c 7.3d 7.3e 7.4a 7.4b Comparison of Results from Different Methods for the Mass of A-pillar to Roof Rail Joint ................................................................................................... 156 Ranges and States of Design Variables .............................................................. 219 Comparison of Optimum Results when Starting from Different Initial Points ............................................................................................................................. 220 Zones of Database ............................................................................................. 221 Comparison of Optimization Results and FEA Results for B-pillar to Rocker Joint (using RSP Translators) ................................................................ 221 Comparison of Optimization Results and FEA Results for B-pillar to Rocker Joint (using NN Translators) ................................................................. 222 Comparison of Correlation Coefficients Obtained using RSP and NN Translators........................................................................................................... 222 States of Design Variables and Optimum Designs for Car A ............................ 223 States of Design Variables and Optimum Designs for Car B ............................ 225 Comparison of FEA Results and Results from Translator B for B-pillar to Rocker Joint ....................................................................................................... 226 Comparison of Mass of the Initial and Optimum Design for Car A and Car B ................................................................................................................. 226 Ranges and States of Design Variables .............................................................. 287 Comparison of Optimum Results when Starting from Different Initial Points ............................................................................................................................. 288 Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint (using RSP Translators) ............................................................................. 288 Values of Design Variables of Four Optimum Designs from RSP Translators........................................................................................................... 289 Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint (using NN Translators) ............................................................................... 290 Values of Design Variables of Four Optimum Designs from NN Translators ... 291 Comparison of Correlation Coefficients Obtained using RSP and NN Translators........................................................................................................... 292 State and Ranges of Design Variables of Car A ................................................ 293 Optimum Design for Car A................................................................................. 294 xii 7.4c 7.4d 7.4e 7.4f States and Ranges of Design Variables of Car B ............................................... 296 Optimum Design for Car B ................................................................................. 297 Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint using RSP and NN Translators................................................................... 298 Comparison of Mass of the Initial and Optimum Designs for Car A and Car B ................................................................................................................. 299 xiii List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 2.6 4.1a 4.1b 4.2 4.3a 4.3b 4.4 4.5 4.6 4.7* 4.8 4.9* Body Structural Joints .......................................................................................... 23 Definition of Stiffness for B-pillar to Rocker Joint ............................................ 24 Definition of Stiffness for A-pillar to Roof Rail Joint ......................................... 25 Architecture of a Typical Multi-layer Neural Network ........................................ 26 Comparison Between Structural Analysis and Neural Network Simulation ........ 27 B-pillar to Rocker Joint ......................................................................................... 28 A-pillar to Roof Rail Joint .................................................................................... 29 Method for Creating Translator A ........................................................................ 53 Method for Developing Translator A using Second Degree Polynomials ........... 54 Comparison of Predictions from Linear Polynomial Model and FEA Results and Explanation of Double Regression ................................................................ 55 Architecture of a Typical Multi-layer Neural Network ....................................... 56 Comparison of Two Different Normalization Methods........................................ 57 Relation Between Standard Deviation and Time of Training .............................. 58 B-pillar to Rocker Joint ........................................................................................ 93 Definition of Stiffness for B-pillar to Rocker Joint .............................................. 94 Parts of B-pillar to Rocker Joint ........................................................................... 95 Extended B-pillar Reinforcement ......................................................................... 96 Non-Extended B-pillar Reinforcement ................................................................ 97 Pillar Bridge Reinforcement ................................................................................. 98 Bulkhead Reinforcement ...................................................................................... 99 Two Different Types of Rocker Cross Section .................................................. 100 B-pillar Orientation ............................................................................................ 101 Rocker Cross Section ......................................................................................... 102 B-pillar Dimensions ........................................................................................... 103 4.10* B-pillar to Rocker Blending Radii and Rocker Dimensions .............................. 104 4.11* Extended Pillar Reinforcement .......................................................................... 105 4.12* Opening in Back of Pillar ................................................................................... 106 xiv 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 5.1a 5.1b 5.2 5.3 5.4 5.5 5.6 Flanges and Spot Welds ..................................................................................... 107 Dimensions for Bulkhead ................................................................................... 108 Dimensions for Back Rocker and Pillar Reinforcement .................................... 109 FEA Mesh of B-pillar to Rocker Joint ............................................................... 110 Comparison of Polynomial and Neural Network Results for I/O Stiffness ........ 111 Comparison of Polynomial and Neural Network Results for F/A Stiffness ...... 112 Comparison of Polynomial and Neural Network Results for Torsion Stiffness .............................................................................................................. 113 Comparison of Polynomial and Neural Network Results for Mass ................... 114 Comparison of Two Different Normalization Methods ..................................... 115 Relation Between Standard Deviation and Time of Training ............................ 116 Relation Between Standard Deviation and Number of Neurons......................... 117 Comparison of FEA Results and Predictions of RSP and NN Translators for the I/O Stiffness of B-pillar to Rocker Joint ....................................................... 118 Relation Between Cp and p for I/O Stiffness ..................................................... 119 Relation Between Cp and p for F/A Stiffness .................................................... 120 Relation Between Cp and p for Torsion Stiffness .............................................. 121 Relation Between Cp and p for Mass ................................................................. 122 Relation Between AIC and p for Different Regression Models of I/O Stiffness .............................................................................................................. 123 Relation Between AIC and p for Different Regression Models of F/A Stiffness .............................................................................................................. 124 Relation Between AIC and p for Different Regression Models of Torsion Stiffness .............................................................................................................. 125 Relation Between AIC and p for Different Regression Models of Mass ........... 126 A-pillar to Roof Rail Joint .................................................................................. 157 Definition of Stiffness for A-pillar to Roof Rail Joint ....................................... 158 Parts of A-pillar to Roof Rail Joint .................................................................... 159 Parts of A-pillar to Roof Rail Joint .................................................................... 160 A-pillar to Roof Rail Joint: Branch .................................................................... 161 Orientation and Position of Each Branch ........................................................... 162 A-pillar to Roof Rail Joint Parameters ............................................................... 163 xv 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 Global Dimensions ............................................................................................. 164 Physical Parameters of Header ........................................................................... 165 Physical Parameters of Roof Rail ....................................................................... 166 Physical Parameters of A-pillar .......................................................................... 167 Physical Parameters for A-pillar Reinforcement ............................................... 168 Physical Parameters for Part 2 ........................................................................... 169 Physical Parameters for Part 3 ........................................................................... 170 Physical Parameters for Part 4 ........................................................................... 171 Physical Parameters for Part 7 ........................................................................... 172 FEA Model for A-pillar to Roof Rail Joint ........................................................ 173 Correlation Between Torsion Stiffness and the Value of (Thickness of Part 3/A_pillar_offset) for A-pillar to Roof Rail Joint .............................................. 174 Comparison of FEA Results and Predictions from Linear Polynomial Model ............................................................................................................................. 175 Comparison of FEA Results and Predictions from Second Degree Polynomial Models ............................................................................................ 176 5.20a Relation Between FEA Results and Error of Prediction for the Linear Polynomial Model of I/O Stiffness ..................................................................... 177 5.20b Comparison of FEA Results and Prediction from Linear Polynomial Model and FEA Results and Explanation of Double Regression .................................. 178 5.21a Comparison of Results for I/O Stiffness of A-pillar to Roof Rail Joint ............. 179 5.21b Comparison of Results for F/A Stiffness of A-pillar to Roof Rail Joint ............ 180 5.21c Comparison of Results for Torsion Stiffness of A-pillar to Roof Rail Joint ..... 181 5.21d Comparison of Results for Mass of A-pillar to Roof Rail Joint ........................ 182 5.22a Scatter Plots for the Fitting Results of Polynomial Translators ......................... 183 5.22b Scatter Plots for the Testing Results of Polynomial Translators ........................ 184 5.22c Scatter Plots for the Training Results of Neural Network Translators .............. 185 5.22d Scatter Plots for the Testing Results of Neural Network Translators ................ 186 5.23a Validating using Cp Criterion for I/O Stiffness ................................................. 187 5.23b Validating using Cp Criterion for F/A Stiffness ................................................ 188 5.23c Validating using Cp Criterion for Torsion Stiffness .......................................... 189 5.23d Validating using Cp Criterion for Mass ............................................................. 190 xvi 5.24a Validating using AIC Criterion for the I/O Stiffness of A-pillar to Roof Rail Joint ..................................................................................................................... 191 5.24b Validating using AIC Criterion for the F/A Stiffness of A-pillar to Roof Rail Joint ........................................................................................................... 192 5.24c Validating using AIC Criterion for the Torsion Stiffness of A-pillar to Roof Rail Joint ............................................................................................................ 193 5.24d Validating using AIC Criterion for the Mass of A-pillar to Roof Rail Joint ..... 194 6.1 6.2 6.3 6.4 6.5 6.6 6.7a 6.7b 6.8a 6.8b 6.9 Lengths of B-pillar Branch.................................................................................. 227 Constraints on the Pillar Reinforcement ............................................................ 228 Manufacturing Constraints on the Lengths of Rocker Cross Section ................ 229 Manufacturing Constraints on Angles of Rocker Cross Section ........................ 230 Constraints on the Rear Plate of Rocker and Angle of Outer Rocker Shell........ 231 Spring Back Angles on Rocker Cross Section ................................................... 232 Explanation of Die Lock Angle .......................................................................... 233 Die Lock Angles on the Rocker Cross Section .................................................. 234 Explanation of Draw Angle................................................................................. 235 Constraints on Draw Angle and Draw Steps....................................................... 236 Constraint on the Cross Section of Rocker ......................................................... 237 6.10a Correlation Between I/O and F/A Stiffness for Designs in the Database ........... 238 6.10b Correlation Between I/O and Torsion Stiffness for Designs in the Database..... 239 6.11a Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the I/O Stiffness of B-pillar to Rocker Joint ...................... 240 6.11b Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the F/A Stiffness of B-pillar to Rocker Joint ..................... 241 6.11c Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the Torsion Stiffness of B-pillar to Rocker Joint .............. 242 6.11d Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the Mass of B-pillar to Rocker Joint ................................. 243 6.12a Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the I/O Stiffness of B-pillar to Rocker Joint ....................... 244 6.12b Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the F/A Stiffness of B-pillar to Rocker Joint ...................... 245 xvii 6.12c Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the Torsion Stiffness of B-pillar to Rocker Joint ................ 246 6.12d Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the Mass of B-pillar to Rocker Joint ................................... 247 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 7.1 7.2 Comparison of Rocker Cross Sections of the Initial Design and Optimum Design.................................................................................................................. 248 Relation Between Mass and I/O Stiffness Requirements for B-pillar to Rocker Joint ....................................................................................................... 249 Relation Between Mass and F/A Stiffness Requirements for B-pillar to Rocker Joint ....................................................................................................... 250 Relation Between Mass and Torsion Stiffness Requirements for B-pillar to Rocker Joint ....................................................................................................... 251 Relation Between the Lower Bound of Thickness of Front Rocker and the Mass of Optimum Design of B-pillar to Rocker Joint ....................................... 252 Relation Between the Lower Bound of Thickness of Pillar Back and the Mass of Optimum Design of B-pillar to Rocker Joint ....................................... 253 Relation Between the Lower Bound of Pillar_base and the Mass of Optimum Design of B-pillar to Rocker Joint ..................................................... 254 Relation Between the Upper Bound of Pillar_base and the Mass of Optimum Design of B-pillar to Rocker Joint ...................................................... 255 Relation Between the Upper Bound of Outer_pillar_width and the Mass of Optimum Design of B-pillar to Rocker Joint ...................................................... 256 Relation Between the Upper Bound of Pillar_inner_length and the Mass of Optimum Design of B-pillar to Rocker Joint ..................................................... 257 Relation Between the Lower Bound of Door_edge_width and the Mass of Optimum Design of B-pillar to Rocker Joint ..................................................... 258 Relation Between the Lower Bound of Rocker_width and the Mass of Optimum Design of B-pillar to Rocker Joint ..................................................... 259 Relation Between the Lower Bound of Outboard_cell_width and the Mass of Optimum Design of B-pillar to Rocker Joint ................................................. 260 Definition of Lengths of Three Branches for A-pillar to Roof Rail Joint........... 300 Constraints on Dimensions of Roof Rail Cross Section ..................................... 301 xviii 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 Constraints on Angles of Roof Rail Cross Section ............................................. 302 Constraints on Dimensions of A-pillar Cross Section ........................................ 303 Constraints on the Angles of A-pillar Cross Section .......................................... 304 Constraints on A-pillar Reinforcement ............................................................... 305 Constraints on the Inner Plate of A-pillar ........................................................... 306 Explanation of Spring Back Angle...................................................................... 307 Manufacturing Constraints on Lengths of Cross Section of Header................... 308 Manufacturing Constraints on Angles of Cross Section of Header .................... 309 Manufacturing Constraints on Lengths of Cross Section of Roof Rail............... 310 Manufacturing Constraints on Angles of Cross Section of Roof Rail ................ 311 Constraints on the Spring Back Angles and Die Lock Angle of Roof Rail Cross Section....................................................................................................... 312 Manufacturing Constraints on Lengths of A-pillar Section ................................ 313 Manufacturing Constraints on Angles of the Outer Shell of A-pillar Cross Section................................................................................................................. 314 Manufacturing Constraints on Angles of A-pillar Reinforcement...................... 315 Constraints on Spring Back Angels and Die Lock Angles of A-pillar Cross Section................................................................................................................. 316 Styling Constraint................................................................................................ 317 Explanation of Safety Constraints....................................................................... 318 Constraint on an Angles of Roof Rail Cross Section .......................................... 319 Constraint on A-pillar Blending Radius.............................................................. 320 Translator for the I/O Stiffness of A-pillar to Roof Rail Joint ............................ 321 7.22a Comparison of FEA Results and Optimization Results Obtained using RSP 7.22b Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the F/A Stiffness of A-pillar to Roof Rail Joint ........................... 322 7.22c Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the Torsion Stiffness of A-pillar to Roof Rail Joint..................... 323 7.22d Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the Mass of A-pillar to Roof Rail Joint........................................ 324 7.23a Comparison of FEA Results and Optimization Results Obtained using NN Translator for the I/O Stiffness of A-pillar to Roof Rail Joint ............................ 325 xix 7.23b Comparison of FEA Results and Optimization Results Obtained using NN Translator for the F/A Stiffness of A-pillar to Roof Rail Joint ........................... 326 7.23c Comparison of FEA Results and Optimization Results Obtained using NN Translator for the Torsion Stiffness of A-pillar to Roof Rail Joint..................... 327 7.23d Comparison of FEA Results and Optimization Results Obtained using NN Translator for the Mass of A-pillar to Roof Rail Joint........................................ 328 7.24 7.25 7.26 7.27 7.28 7.29 7.30 7.31 7.32 7.33 7.34 7.35 7.36 7.37 7.38 7.39 7.40 7.41 Comparison of A-pillar Cross Sections of Initial and Optimum Designs........... 329 Mass of Optimum Design vs I/O Stiffness Requirement.................................... 330 Mass of Optimum Design vs F/A Stiffness Requirement ................................... 331 Mass of Optimum Design vs Torsion Stiffness Requirement............................. 332 Mass of Optimum Design vs H_offset ............................................................... 333 Mass of Optimum Design vs Length of Roof Rail Branch ................................. 334 Mass of Optimum Design vs Length of A-pillar Branch .................................... 335 Mass of Optimum Design vs Angle Theta .......................................................... 336 Mass of Optimum Design vs Angle Phi.............................................................. 337 Mass of Optimum Design vs Thickness of Part 1............................................... 338 Mass of Optimum Design vs Thickness of Part 2............................................... 339 Mass of Optimum Design vs Thickness of Part 3............................................... 340 Mass of Optimum Design vs Thickness of Part 7............................................... 341 Mass of Optimum Design vs the Lower Bound of RR_width ............................ 342 Mass of Optimum Design vs the Lower Bound of Door_allowance .................. 343 Mass of Optimum Design vs the Upper Bound of Door_allowance................... 344 Mass of Optimum Design vs the Lower Bound of H_width............................... 345 Mass of Optimum Design vs the Lower Bound of AP_blending_rad ................ 346 xx Nomenclature AIC bik ci cov CAD CAM i CN CP DOE DOF Ed En f i (⋅) F( ⋅ ) F/A FEA FEM g j (⋅) h(⋅) I/O IMSE KD KF / A ˆ K F/A k Akaike’s Information Criterion bias of the ith neuron in kth layer in a neural network coefficient in a polynomial coefficient of variation computer aided design computer aided manufacturing combination of i number from a total of N numbers Mallows criterion design of experiment degree of freedom d-dimension space n-dimension space transfer function of a neuron objective function forward/afterward finite element analysis finite element method inequality constraint equality constraint inboard/outboard Integrated Mean Squared Error prediction from a double regression model, can be I/O, F/A, torsion stiffness or mass predicted F/A stiffness target F/A stiffness value used to normalize F/A stiffness predicted I/O stiffness target I/O stiffness KF / A K I /O ˆ K I /O xxi K I /O KL KQ K Tor ˆ K Tor value used to normalize I/O stiffness prediction from a linear polynomial model, can be I/O, F/A, torsion stiffness or mass prediction from a quadratic polynomial model, can be I/O, F/A, torsion stiffness or mass predicted torsion stiffness target torsion stiffness value used to normalize torsion stiffness mass modified feasible direction method massively parallel processing number of designs number of designs used to validate the translator B degree of double regression model number of design variables in regression number of designs used in fitting/training total number of weights and bias of a neural network number of inputs for the ith neuron in the kth layer of a neural network number of constraints in optimization number of design variables in optimization neural networks non-uniform rational B-splines optimal experimental design number of terms in a polynomial average of ri ratio of predicted value over measured value K Tor M MFD MPP n nC nD nV nf nm n k n ,i NCON NDV NN NURBS OED p r ri R 2 coefficient of determination (R-square) response surface polynomials square root of estimate of variance of the error in the measurements search direction in optimization simulated annealing sequential linear programming RSP s Sq SA SLP xxii SQP SSE SSR SSTot wik, j xi xiL xiU X y FEA , j yi yO , j y FEA yi yO ˆ yD ˆ yi ˆ yL YFEA YO zik sequential quadratic programming sum of square errors regression sum of squares total sum of squares of the variations of measurements from the mean weight of ith neuron in the kth layer of a neural network on the input from the jth neuron of (k-1)th layer ith variable lower bound of the ith design variable upper bound of the ith design variable vector of variables FEA result of jth design measured response optimization result of jth design average of FEA results average response average of optimization results prediction of response from double regression model prediction of response prediction of response from linear polynomial vector of the FEA results vector of optimization results total input for the ith neuron in the kth layer coefficient in objective function of translator B step used in optimization coefficient used in optimization correlation coefficient standard deviation coefficient used in sigmoid function multiplier in sequential linear programming α λ β ρ X ,Y σ θ ϑ xxiii Chapter 1 Introduction This chapter defines the problem of developing approximate design tools for joints in Section 1.1. Section 1.2 reviews previous studies on analysis and design of joints in automotive structures and approximate tools for rapid analysis of structures. Section 1.3 explains the objectives and benefits of the project presented in this dissertation. 1.1 Definition and Significance of the Problem This section defines the problem of developing approximate tools for rapid analysis and design of automotive joints, and explains why these tools are important. Then it presents the goals of the study presented in this dissertation. Typically, design of a complex system starts by setting targets for its performance characteristics. Then, design engineers cascade these targets to the components and design the components to meet these targets. A hierarchical approach is followed in design of many real life complex systems, such as cars: first, the overall system is optimized using a concept model, which represents the constituent components using engineering parameters. Then, the components are optimized. For example, in design of a car body, first, we optimize the values of the engineering parameters describing the performance characteristics of its components (e.g., cross sectional properties of beams, and stiffness of joints). Then, we optimize the design variables of the components, such as plate gages and blending curve radii, using the optimum values of the engineering parameters. This approach allows design engineers to work with relatively simple models, which involve a manageable number of parameters, at each level. 1 Joints affect significantly the static and dynamic behavior of a car (Chang, 1974). When using the hierarchical approach we need tools that, for a given set of values of the engineering parameters of a joint, rapidly determine if there is a feasible design that corresponds to these values and estimate the mass and dimensions of this design. Traditionally, a joint is defined to be any region of a structure containing the intersection or junction of two of more load-carrying members. A joint is formed by metal sheets fastened by spot welds. In the automotive industry, seven regions of the car body are generally considered to be joints, namely, A-pillar to roof rail, A-pillar to hinge pillar to shotgun to cowl, B-pillar to roof rail, C-pillar to roof rail, front hinge pillar to rocker, B-pillar to rocker, and rocker to rear quarter (Figure 1.1). Joints are flexible; they allow relative rotation between the adjacent branches. The flexibility of joints affects significantly the static and dynamic response of car bodies. In concept models of car bodies, a joint is modeled by allowing its most flexible branch to rotate relative to the remaining branches, which are rigidly connected. The performance of a joint is characterized by its stiffness and mass. To define properly the stiffness, one should use the complete stiffness matrix corresponding to the ends of the joint branches. For example, the performance of the joint in Fig. 1.2 should be characterized by an 18×18 stiffness matrix corresponding to the points A, B, C at the ends of the three branches. However, this would require use of a large number of elements to characterize the performance of a joint. For example, we would need 78 elements to completely specify the stiffness matrix of the B-pillar to rocker joint in Fig. 1.2. Therefore, automotive manufacturers use simpler models involving fewer parameters to characterize the performance of a joint. A joint is thought as a set of beams connected by hinges at the joint center. Rotational springs constraint the rotations of the members relative to each other. The stiffnesses of these springs are used to characterize the performance of a joint. Usually, all branches are assumed rigidly connected except for the most flexible one. The stiffness of a spring in a particular direction is estimated by applying a moment to the most flexible branch and dividing this by the resulting rotation of the branch in the direction of the moment. Note that since there is coupling between the stiffnesses in 2 different directions, the stiffness obtained using above method may be considerably different than the corresponding element in the stiffness matrix of the joint. Figure 1.2 shows the definition of three stiffnesses, namely, inboard/outboard (I/O), forward/afterward (F/A) and torsion stiffnesses, for the B-pillar to rocker joint. Figure 1.3 explains the definition of these stiffnesses for A-pillar to roof rail joint. Here I/O, F/A and torsion stiffnesses are defined by Moment in a direction Resulting rotation in same direction Stiffness = In the preliminary design stage of a car, it is important to have simple, designoriented tools that can predict the stiffness and mass of a joint rapidly once its dimensions and type of construction are specified. Another type of tool is needed that finds a feasible design, that is a design that satisfies packaging, manufacturing and styling constraints if the stiffness and mass requirements are specified. Unfortunately, in many industries, such as the automotive, there are no such tools. Instead, crude empirical relations are used to link engineering parameters to physical design variables and mass, which can lead to setting unrealizable targets for the engineering parameters of the components. It is not efficient to use Finite Element Analysis (FEA) to predict the performance of a joint because we do not have enough information to model joints in the concept design stage, and FEA could be too slow. One way to overcome this problem is to develop approximate tools that link engineering parameters to physical design variables. These tools should be efficient so that they can be used to perform many design iterations at low cost. We call these tools translators. The first tool (Translator A) can translate information about the parameters that define the geometry of a joint into information about the stiffness and mass. The second tool (Translator B) can translate information about the requirements of the stiffness and/or the mass of a joint into information about design variables. Response surface polynomials (RSP) and neural networks (NN) can be used to develop such approximate tools because they can greatly reduce the time required to analyze a design. The goals of the study presented in this dissertation are to develop: a) a methodology for developing translators A and B for a joint using neural networks and response surface 3 polynomials. b) demonstrate and validate the methodology on some actual automotive joints. c) compare neural networks and response surface polynomials. Translator A rapidly calculates the stiffness and mass of a joint whose dimensions are specified. Translator B solves the inverse problem; given some requirements about the stiffness, it finds an optimum design of the joint that satisfies these requirements as well as given packaging, styling and manufacturing constraints. 1.2 Previous Work This section reviews studies in the following areas that are important to this dissertation: a) Joints in automotive structures. b) Response surface polynomials and neural networks in structural analysis and design. c) Optimization of automotive structures. d) Software for computer aided design (CAD). 1.2.1 Structural Analysis of Joints in Automotive Structures When engineers started using FEA for analysis and design of car bodies, they recognized that it is incorrect to consider a joint as rigid, that is, to assume that one joint branch does not rotate relative to the other branches. Chang (1974) demonstrated that a car body frame model that assumes that joints are rigid underestimates seriously static deflections. Therefore, the flexibility of joints must be taken into account in order to predict accurately the static and dynamic response of a car. Since 1970's researchers have tried to understand the deformation mechanisms of joints and to develop simplified models for joints. Chang (1974) studied the influence of body connection flexibility on the in-plane response of an automobile body structure. He used an element consisting of springs and hinges to account for the joint flexibility in the analysis of the overall car body stiffness. Nikolaidis and Lee (1992) developed a procedure to estimate the parameters of simplified joint models that consist of springs using FEA results of very detailed models of joints or from measurements. Both studies 4 demonstrated that a frame model of the car body will be more accurate if it accounts for the joint flexibility. Sunami and Yoshida (1987,1990) studied the stiffness of T- and L- shaped joints made of two box beams under in-plane and out-of-plane bending, respectively. They used plates and stringers to model the joints and obtained analytical expressions of joint deflections using shear flow theory. They also studied the effects of bulkhead on the rigidity of joints. Their results agreed well with both experimental and FEA results. They found that inserting bulkheads is an effective means of preventing cross-sectional deformation, and thus increases the joint rigidity. However, these approaches have limited application because they apply to rectangular cross-sections only and they can not account for details in the construction of a joint, such as welding access holes or spot welds. Zhu (1994) developed a generic CAD model for the B-pillar to rocker joint. In creating the model, he used flat plates to model the blending areas of the joint. This model predicted accurately the inboard/outboard (I/O) stiffness. However, this model did not predict the forward/afterward (F/A) stiffness and the torsion stiffness accurately (see Figure 1.2 for the definition of I/O, F/A, and torsion stiffness). The relative error for F/A and torsion stiffness exceeded 30%. Zhu’s model was relatively simple and it applied to one type of construction and one shape only. Specifically, his model could not account for different types of reinforcements, different types of cross sections, and details in the geometry of a joint. 1.2.2 Response Surface Polynomials in Structural Analysis and Design Using response surface polynomials (RSP) to model the performance, mass and cost of a system is a mature technique. Usually, linear or quadratic polynomials are used. RSP become prohibitively expensive when cubic and higher degree polynomials are chosen for design problems involving several variables. In addition, cubic and higher degree polynomial models may contain one or more inflection points. In gradient-based numerical optimization schemes the optimizer may converge to an inflection point rather than to a local or global optimum (Giunta, 1997). 5 RSP have been widely used. For example, Giunta (1997) applied RSP to the analysis and design of aircraft. He used stepwise regression to obtain the best model. Balabanov (1997) used quadratic polynomials to construct a response surface polynomial for the weight of the wing bending material. In his study, he used backward stepwise regression to obtain the best regression model. Venter et al. (1996) discussed a procedure for constructing response surface approximations for design analysis and optimization. In this paper, Venter et al. used a response surface to approximate the response of a homogeneous isotropic plate, which is a function of the thickness. The fitting and testing data were obtained using FEA. Mixed stepwise regression (a procedure employing both forward and backward stepwise regressions) was used in this paper to obtain the best regression model. The developed response surfaces were very accurate (the error was about 3%). Roux et al. (1996) reviewed and investigated techniques for finding more accurate approximations, including, experimental design techniques, selection of the “best” regression equation, intermediate response functions and the location and size of the region of interest. An intermediate response function uses a function of design variables as its independent variable. In this paper, they considered three examples, including a two-bar truss, a three-bar truss, and a ten-bar truss. They found that the use of engineering knowledge about the problem could yield large benefits in terms of accuracy of the approximation and size of the region over which it is applicable. If higher accuracy is desired for a given approximation, it is best achieved by using smaller subregions – thereby reducing the bias error. The use of more experimental points could make the response surface more accurate over the region of interest, but the ultimate accuracy is limited by the choice of the approximating function. The bias error (error due to lack of fit) cannot be eliminated through the use of more experimental points. If a low order function (e.g., first or second degree polynomial) is used on data having a very small amount of noise, the selection of the “best” regression equation does not significantly contribute to the optimization process. 6 Zhu (1994) built a translator for an automotive joint using RSP. In constructing the RSP translator, he used a complete second degree polynomial. When increasing the number of design variables defining the geometry of a joint in the polynomial, the generalization performance of the polynomial translator first improved, but after a certain number, it gradually deteriorated. Generalization performance is the capability of an approximation tool to predict the stiffness and mass of designs that it did not see during fitting. The reason was that Zhu used a complete second degree polynomial that included some unimportant terms. The coefficients of these terms were not estimated accurately, which reduced the generalization performance. Although Zhu’s approach yields reasonably accurate approximating polynomials in most cases, the approach requires that the user identifies and excludes unimportant terms. 1.2.3 Neural Networks in Structural Analysis and Design 1.2.3.1 Neural Networks The area of neural networks (NN) is relatively new. Recently, it has attracted a lot of attention from both researchers and practitioners. Figure 1.4 shows a typical multi-layer feed forward NN. A NN is composed of computational elements called neurons. The neurons are generally arranged in several layers. The function that transfers the input information of a neuron into the output information is called transfer function. There are several different transfer functions, such as the sigmoid function, and linear function. These functions will be discussed in detail in the second chapter. We generally present some examples (exemplars) to a NN, from which the NN generates general information through a training process. There are different learning algorithms to train a NN. NN learning is performed mainly through the readjustment of the weights through the application of certain learning algorithms. NN can be classified as either supervised or unsupervised according to the way by which weights are adjusted. A supervised NN will require information to be provided both from the NN itself (local information), and from the environment (external information) to guide its learning task. On the other hand, for an unsupervised NN, the learning process is accomplished based 7 on the local information only, implying that the NN involved must be self-adaptive without the help provided by external guidance. Back-propagation is a learning algorithm for supervised NN. Competitive learning is a learning algorithm for unsupervised NN. The competitive learning scheme provides a typical example of unsupervised NN that modify their parameters during training without external guidance (Wu, 1992). A number of different NN have been proposed. Different training algorithms have been studied by Lippmann (1987), and Alon et al. (1991). Huang et al. (1991) investigated some fundamental issues concerning the capability of multi-layer perceptrons with one hidden layer. The perceptron was first conceived by Rosenblatt in 1958. The input of a perceptron is a n-dimensional vector. The perceptron forms a weighed sum of the n components of the input vector and adds a bias value. A perceptron is the same as a neuron in this study. Huang et al. (1991) focused on realizations of functions that map from a finite subset of a n-dimension space E n into a ddimension space E d . Both real-valued and binary-valued functions were considered. They derived a least upper bound for the number of hidden neurons needed to realize an arbitrary function that maps from a finite subset of E n into E d . They also obtained a nontrivial lower bound for realizations of injective functions (In this paper, the injective function mapped the inputs in a m-polytope into the outputs into the same m-polytope. Therefore, it is a one-to-one mapping function), which are useful in studying pattern recognition and database retrieval. 1.2.3.2 Application of Neural Network in Structural Analysis Many engineers in several industries have tried to apply NN to practical problems (Jagota, 1994, Hajela, and Berke, 1990). Wu (1992) examined the architecture and functionality of NN, and provided some examples of applications in manufacturing. In this paper, Wu explained the basic concepts in NN, including neurons, transfer functions, classification of neural networks, and learning algorithms. 8 Hajela and Berke (1990) used NN to represent the force-displacement relationship in static structural analysis. The trained NN provided computationally efficient capabilities for reanalysis, and were well suited for optimum design. They used underdetermined NN, that is, networks that had more unknown parameters than the number of training pairs. Keeler (1992) used NN and fuzzy logic to predict and optimize manufacturing processes. He found that NN require little human expertise; the same neural network algorithm will work for many different systems. The predictions of NN are nonlinear functions of the parameters, which means that NN can be more realistic models than RSP in same problems. NN typically work much better than traditional rule-based expert systems on problems in which the important relationships and rules are difficult to discern, or the number of rules can be overwhelming. NN are powerful nonlinear regression algorithms, and have proven to be the ideal tool for leveraging inexpensive computer time to build models directly from data in which the input vector and/or the output vector have many components. NN achieves this by “learning” the underlying relationships between the input data and the output response. Cook and Shannon (1992) used NN to predict the occurrence of out-of-control process conditions in a composite board manufacturing facility. The manufacturing operations considered were complex, involved many process parameters, and the relationships and interactions of these variables were difficult to understand. When applying a multi-variable regression model to these operations, they found that it could explain only 25% of the variation in the response variable. However, the values of many process variables were typically collected by the operators, allowing data sets of examples of operating conditions to be constructed. Cook and Shannon used backpropagation to train NN to represent the values of the process parameters along with historical data on relevant parameters, including temperature, moisture content, and bulk density. The trained NN were able to successfully predict the state of control of the specific manufacturing process parameters with 70% accuracy, thus, demonstrating the potential of NN in manufacturing process analysis. In this paper, they also observed that 9 some NN were overtrained. However, they did not propose any method to avoid this problem. Since it generally takes too long to train a NN, some researchers tried to train the NN in a massively parallel processing (MPP) environment (Shieh, 1994). In his study, Shieh used a 4-layer NN to simulate the structural analysis process of trusses with 10 to 1008 bars. Shieh performed a large number of structural response analyses that is required for training a set of relatively large size NN very efficiently in a single run in parallel in a MPP environment. The trained NN provided a nearly instant output solution from a given set of input variables. A lot of work has been done on the application of NN in optimization (Berke and Hajela, 1992). There are mainly two approaches of using NN in the optimization process. The first approach creates a NN model of the structural response and then attaches it to any conventional optimization algorithm. In the second approach, a NN model is trained using optimum designs that correspond to different requirements and conditions. The NN is trained to estimate the optimum design directly for given design conditions and bypass all the analyses and optimization iterations of the conventional approach. In many practical problems, successful similar designs could be collected within some domain of design conditions. These designs can be used as input-output pairs to train a NN that can serve as “intelligent corporate memory”. This network can provide a new design for different design requirements. Swift and Batill (1991) demonstrated the use of NN for preliminary structural design. In their study, the authors used NN to simulate the structural analysis process of a 5-bar truss, a 10-bar truss, and a wing box. Using NN as a fast structural analysis tool, the authors optimized the structures considered. A back-propagation training algorithm was used. The number of unknown parameters in the NN was larger than the total number of training designs (i.e. the NN used was underdetermined). They used a pre-specified percentage error as a target and stopped training process when this target was met. The NN worked well in the examples considered in this paper. However, an underdetermined NN may have poor generalization performance in other applications. 10 Botkin and Lust (1995) applied NN to the shape optimization of some automotive components, including an upper control arm and a steering knuckle. They used a multilayer feed forward NN as a design tool during preliminary design to approximate the mapping relation of the optimal design space. They investigated the effect of various combinations of key part dimensions on the optimal part mass using parametric studies. Since the training data was relative expensive to generate, the NN was trained using a relatively small amount of training data (36 input-output pairs) evenly distributed over the optimal design space of interest. Near optimal values of the design variables, performance measures and the objective function (typical mass) were obtained as outputs from the networks. The trained NN could predict the optimal part mass, design variable values and performance measures with very small error (3.5%). Zhu (1994) built a translator using both RSP and NN. In obtaining the NN results, he used the Box-Behnken method to create the training pairs. He found that the NN translator gives better predictions than the RSP translator. Cohn (1994) used optimal experimental design (OED) to create the input-output mappings for training. OED is concerned with the design of experiments that are expected to minimize variability in a parameterized model. The intent of OED is usually to maximize confidence in a given model, minimize parameter variances for system identification, or minimize the model’s output variance. An experimental design method allow us to efficiently explore the input space, which gives us the most amount of information using the smallest number of training sets. In structural design of aerospace or automotive vehicles, the difficulty of choosing among a set of discrete design variables must be faced. Conventional optimization methodologies, which are typically based on sensitivity analysis and gradient based search techniques, are not suitable for the discrete problem. Optimization problems dealing with discrete design variables, while often mathematically well defined, are typically difficult to solve in practice. Swift and Batill (1992) applied NN to the optimization of structures involving discrete design variables using a simulated annealing 11 (SA) algorithm. Simulated annealing, which is based on an analogy to the annealing process in metallurgy, has been used effectively for combinatorial optimization problems. In this paper, they used two trusses and a wing as examples to test the proposed method. Objective function was the mass for the three examples. Design variables for the first two examples were the types of materials of the rods in the truss, and the lamina orientations for the wing example. NN were used to approximate the relation between the design variables and the response. They found that the application of SA and NN appears promising for the design problems considered. 1.2.3.3 Effects of Disturbance and Cross-validation on NN Since there is generally some noise in the training data, some work has been done on the effects of the corrupted data and the ways to improve the robustness of NN. Liu (1994) studied the robustness of a conventional back-propagation algorithm for NN regression. He found that this algorithm is robust to leverages (data with x corrupted), but not to outliers (data with y corrupted). Adding a regularizer to the error function being minimized during training is a technique for preventing a NN from overfitting noisy data. Regularizer is a function, such as a spline that can typically smooth the fit to noisy data. Wu and Moody (1995) derived a smoothing regularizer for general dynamic models, which was derived by considering perturbations of the training data. They also tested the regularizer in several case studies, and found that it performed well. Tresp et al. (1994) studied how to incorporate data with uncertain or missing information into the training of NN. In their work, the general solutions required a weighed integration over the range of unknown or uncertain inputs. Many researchers noticed that the generalization error decreases in an early period of training, reaches the minimum and then increases as training continues, whereas the error of training results monotonically decreases. Generalization error is the error of a polynomial or a neural network when using this tool to predict the performance of designs that were not in the training or fitting set. Therefore, it is advantageous to stop 12 training at a certain point in time. Amari et al. (1995) studied the effect of crossvalidation on the generalization error of the trained NN. To avoid overtraining, the following stopping rule has been proposed, which is based on cross-validation. Divide all the available examples into two sets. One set is used for training. The other set is used for testing in a way that the behavior of the trained network is evaluated by using the test examples and training is stopped at the point that minimizes the testing error. They found that when the number nm of network parameters is large, the best strategy is to use almost all n f examples in the training set and to use only 1 / 2nm percent examples in the testing set. For example, when nm =100, only 7% of the training patterns are to be used in the set determining when to stop training. When n f >30 nm , the results show that cross-validation does not improve the generalization error, so no overtraining is observed. Because in most real actual problems, the number of examples is limited, early stopping does improve significantly the generalization ability of a network. Generalization ability is the capability of a polynomial or a neural network to predict designs that it had not seen before. Plutowski et al. (1994) studied the effects of cross-validation on the estimates of NN. They found that the cross-validation average squared error is an unbiased estimate of the Integrated Mean Squared Error (IMSE). IMSE is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. Dodier (1995) presented a theory about early stopping for linear networks. He found that given a training set and a validation set, all weight vectors found by early stopping must lie on a quadric surface, which is usually an ellipsoid. 1.2.4 Optimization of Automotive Structures Optimization of automotive structures typically needs to evaluate the performance characteristics of a design (stress, strain, or stiffness) many times. This repeated analysis process can be done using FEA or approximate tools such as RSP and NN. A lot of work has been done on the application of optimization techniques to automotive structural design. 13 Botkin (1992) proposed to work with design-oriented geometric features associated to part recognizable features. He integrated a fully automatic mesh generator into the threedimensional shape optimization of a steering knuckle of a car. Unlike other papers on shape optimization, in which stress and strain constraints are imposed on every finite element and the number of elements remains constant throughout the iteration history, he used an automatic mesh generator that allowed the number of elements to vary throughout the convergence history. The stress and strain constraints were assigned to the design model rather than to the mesh. Lindby and Santos (1994) proposed an approach that uses the capabilities offered by CAD modelers to provide shape design parameterization and design velocity computations, and uses a commercial finite element code for performance analysis. They used the shape optimization of a torque arm to illustrate the proposed approach. Schramm et al. (1993,1994) used non-uniform rational B-splines (NURBS) in shape optimization of a thin walled beam. They used NURBS curves to describe the cross sectional shape of the beam. Design variables were the control parameters that define NURBS curves and the wall thickness of the beam. An advantage of Schramm’s study is that one does not need to remesh the structure as the dimensions change during optimization. However, the method was demonstrated on structures that were practically two-dimensional. One needs to investigate how practical the method is for complex, three-dimensional structures. Belegundu (1993) proposed a way of using basis shapes to optimize the shape of mechanical components. Basis shapes are the different shapes of the components that a designer wants to optimize and their FEA results. Basis shapes can be generated in a number of ways. For example, the shapes from past designs can be used as basis shapes. By using the basis shapes, Belegundu reduced the optimization problem to finding the right combination of the basis shapes. In this paper, Belegundu applied the proposed method to a torque arm component and a connecting rod. The objective function was the weight of the component. The optimum design reduced the weight of the component by 33-40%. 14 Puttre (1993) discussed about combining CAD and FEA to automate the repetitive tasks associated with design optimization. In this paper, Puttre summarized the work on shape optimization using existing CAD and FEA packages. Puttre recommended that the number of design variables for shape-optimization problems not exceed 20. Insignificant design variables should be removed to make the optimization problems more manageable. Fenyes (1992) optimized thin-wall beam type member by taking into account both structural and manufacturing requirements. His approach can be applied in the early design stages. His results indicated that it is possible to simultaneously consider structural and manufacturing constraints for structures composed of simple closed channel sections. However, his results could not be extended directly to complex multi-piece sections, which are typically used for the frame members in a vehicle because the simple formulae he used to evaluate the forming constraints of sheet metals were not applicable to other cross sections or forming conditions. 1.2.5 CAD Software Today, there are many powerful commercial CAD software packages that can help automate the design process. Among all the CAD/CAM programs, I-Deas, Pro/Engineer, CATIA, and PATRAN are among the most popular. CATIA is a CAD/CAM software that was first developed by the French aerospace industry. CATIA has many specific modules and many capabilities. Although PATRAN has many capabilities, it only has limited parametric modeling capabilities. Most people use PATRAN as a pro- or postprocessor of FEA. The author only had access to I-Deas and Pro/Engineer. According to the author’s experience, it is easier to change or redefine the shape or properties of a structure in Pro/Engineer than in I-Deas. I-Deas also lacks some assembly capabilities of Pro/Engineer. It is also easier to develop complex relations between the dimensions of one or more parts in Pro/Engineer than in I-Deas. On the other hand, I-Deas has more function modules, such as more finite element types. Both I-Deas and Pro/Engineer have structural analysis modules. I-Deas also has a structural optimization module. To 15 use the optimization module of I-Deas, the user only needs to create a model in I-Deas, and specify the objective function and constraints. Then, I-Deas can optimize the design. The structural analysis and structural optimization capabilities of I-Deas and Pro/Engineer have greatly improved in the past few years. However, their structural analysis and structural optimization modules are still not superior in speed, accuracy, reliability, and versatility to other special structural analysis codes such as MSC/NASTRAN and ANSYS, and structural optimization codes such as DOT and GENESIS. Moreover, CAD/CAM software such as Pro/Engineer can not create some types of elements, such as rigid elements that are frequently used to simulate the spot welds. 1.2.6 Neural Networks or Polynomials versus Finite Element Analysis in Optimization Structural optimization needs to analyze a large number of alternate designs (from a few dozens to many hundreds). Therefore, the computation effort required for analysis is an important issue. Before performing structural optimization, we need to know whether it is better to use a NN or a polynomial translator to simulate the input-output relations in the optimization process instead of directly using FEA. The following paragraphs compare the cost of using NN and RSP in optimization with the cost of performing FEA in each step of optimization. Neural Networks Developing and verifying a NN requires one to develop a parametric CAD model of the structure, analyze the structure for many combinations of values of the dimensions (from few hundred to few thousand), train and validate the network, and perform optimization. The computational time for the first three tasks is much larger than the last task. Indeed, it takes only a very small fraction of a second to predict the performance of a structure using a NN that has already been trained. Carpenter et al. (1992) compared the efficiency and accuracy of RSP and NN for approximation of functions. In this paper, they used Fox’s banana function, a five bar 16 truss, a 35 bar truss with four design variables, and a 35 bar truss with 15 design variables to compare the performance of NN and RSP. The Fox’s banana function can be 2 expressed as: y = 10 x14 − 20 x2 x12 + 10 x2 + x12 − 2 x1 + 5 . They concluded that the two methods were comparable in terms of accuracy and efficiency. They also suggested that the number of nodes on the hidden layer be somewhere between the average and the sum of the input and output nodes. Shi et al. (1998) compared Holographic NN with RSP. They concluded that Holographic NN are more accurate than RSP. Rogers (1994) compared the time needed by using NN to simulate analysis and the time needed by directly performing structural analysis using FEA in the optimization process. He used a 3000 degrees-of-freedom (DOF) beam as a test problem. The objective was to optimize the shape of the beam to minimize the weight while satisfying stress constraints. He found that there is a break-even point, beyond which, a neural network can reduce the total amount of time required for analysis (Fig. 1.5). Guyot (1996) did a similar comparison between the two approaches. He found that the initial cost to perform optimization using a NN is relatively high. Consequently, if the optimization is performed only once, it may not be advantageous to use a NN to predict the performance of the system because of the high initial cost of developing such a tool. However, the cost of using a NN is negligible once the NN is built. Thus, if we need to run an optimization program many times to perform a parametric study or to try different initial points to find the global optimum, it is advantageous to use a NN as an alternative to FEA in optimization. Polynomials Like NN, RSP can predict the performance characteristics of a design in a fraction of a section, thus, it can greatly reduce the time of optimization. In general, it takes less time to construct and fit a RSP than to construct and train a NN. Therefore, it is better to use a polynomial instead of a NN, if a RSP can be found that can approximate the data with acceptable accuracy. 17 NN and RSP can also be better than FEA if there is noise in the results of structural analysis because they avoid problems with discontinuities in the gradients. A RSP reduces the effects of corrupted data by providing a smooth approximation of the response. Liu (1994) showed that optimization using NN is not sensitive to a few corrupted data. This property of NN will be explained in chapter 2. 1.3 Objectives and Benefits of the Project Presented in this Dissertation The objectives of this dissertation are: a) Develop a methodology for developing translators A and B for design of automotive joints. This methodology consists of the following steps: • Parameterize a joint using a set of few physical design variables that can fully determine the shape of the joint. These design variables can be controlled in the early design stage of a car. • Develop a parametric model. The parametric model should account for packaging, styling and manufacturing constraints, and determine if a design whose design variables and type of construction are specified is feasible. This model should be able to create a FEA data file, which can be used by a finite element analysis code to give reasonably accurate predictions for I/O, F/A, torsion stiffness and mass. • • Develop a database of designs that will be used for developing and validating translator A. Understand and rank the most important design variables using stepwise regression, and identify the most important design variables that affect the performance of the joint most. It is better to use stepwise regression because it introduces only the important parameters to the regression model. • Develop a translator A for design guidance using both RSP and NN, which can predict the performance and mass of a given joint at a fraction of a second. This will allow the use of translator A to optimize the joint, or even the entire car body very efficiently. Unlike Zhu, who used a complete second degree polynomial to construct the polynomial translator, we will identify and 18 only consider the important terms when we construct the polynomial translator. By doing so, we can minimize the effects of unimportant terms on the generalization performance of the translator. • Develop a translator B using both RSP and NN translator A. Translator B can find the values of design variables of a joint that satisfies given performance targets and packaging and manufacturing constraints. This is achieved using optimization, in which we minimize the mass of a joint subject to packaging, manufacturing and styling constraints. b) Apply the methodology to the B-pillar to rocker and A-pillar to roof rail joints of an actual car. For this purpose, we will complete the following tasks: • Parameterize both the B-pillar to rocker joint (Fig. 1.2, and 1.6) and A-pillar to roof rail joint (Fig. 1.3 and 1.7). The reasons that we select B-pillar to rocker and A-pillar to roof rail joints are the following. The B-pillar to rocker joint is the simplest joint in a car, and most cars have similar types of construction. The A-pillar to roof rail joint is more complicated than the B-pillar to rocker joint. We can apply the experience gained from the parameterization of B-pillar to rocker joint and A-pillar to roof rail joint to parameterization of other joints. • Develop two parametric joint models. One is for the B-pillar to rocker joint, and the other is for the A-pillar to roof rail joint. Zhu’s study was a very good step toward developing parametric models of joints. The differences between our B-pillar to rocker joint model and that of Zhu’s (1994) are: − Our models are created using Pro/Engineer and I-Deas. They can account for more design details, such as different reinforcements, spot welds, and access holes. Zhu wrote a FORTRAN program that generated a Finite Element Model given the dimensions of a joint. Straight lines were used in his model to simulate the blending areas. − Our models will account for about seventy packaging, manufacturing and styling constraints. Following are some new constraints that are included in our model. Our 19 model will be able to determine if a joint can be manufactured in terms of stamping. The B-pillar to rocker model will also be able to determine whether the shape of the cross section of the rocker allows the water to run off. The A-pillar to roof rail model will be able to determine whether the A-pillar is too wide so that it obstructs vision, and whether the reinforcement can fit into the two outside parts of A-pillar. − Our model will accurately predict not only I/O stiffness, but also F/A stiffness, torsion stiffness and mass. − The user will only need to change the value of control parameters in order to change the type of construction of a joint. Zhu’s model accounted only for one type of construction. c) Develop a database of example designs for each joint that will be used to fit a response surface polynomial or train a neural network that can predict the performance and mass of a given joint. d) Identify and rank the important design variables for each joint using stepwise regression. e) Develop a translator A for each joint using RSP and NN. Validate and compare the RSP and NN based translators. f) Develop a translator B for B-pillar to rocker joint and A-pillar to roof rail joint. Validate the results of the translator B by comparing the optimization and FEA results, and study the effect of changing the constraints on the most important design variables on the mass of the optimum design. As mentioned in the introduction, this project will establish a methodology for developing two much needed tools for design of joints. The tools will help a designer analyze joints, minimize the weight of the joints of a new car model, and reduce the time it takes to design them. By using this methodology, the designer will find the most important physical design variables (dimensions) that affect the performance of a joint and establish relations between these design variables and packaging and manufacturing 20 constraints, which is very important information in design. The methodology developed in this project will be useful in developing similar design tools for other components of a car such as beams, and local structures at the positions where the suspension is attached to the car body 1.4 Outline of the Dissertation Chapter 2 describes the general methodology for developing translator A. This chapter covers the parameterization of joints, selection of design variables, creation of a database, and development of translators A using RSP and NN. Chapter 3 presents the general methodology for developing translator B, including, selection and classification of design variables, definition of objective function and constraints used in translator B, and validation of translator B. In chapter 4, the general methodology explained in chapter 2 is applied to a B-pillar to rocker joint of an actual car. First, we create a parametric model for the B-pillar to rocker joint. The parametric model can account for packaging, styling, and manufacturing constraints, and determine if a given design whose design variables and type of construction are specified is feasible. These models can create FEA data files, which can be used with a finite element analysis code, such as MSC/NASTRAN, to predict I/O, F/A, torsion stiffness and mass. Using this generic parametric model, translator A is built and validated for the B-pillar to rocker joint. Finally, this chapter compares the RSP and NN based translators. Chapter 5 demonstrates the work on developing translator A on the A-pillar to roof rail joint of an actual car. First, a parametric model is developed. Then, both a polynomial and a neural network translator A are built and validated. The two translators are compared. In Chapter 6, we apply the general methodology for developing translator B, which is explained in chapter 3, to the B-pillar to rocker joint of an actual car to find designs that 21 meet given performance requirements. The objective function and the constraints used in this translator are explained in detail. Then, the translator B is validated. Specifically, we first check if the translator B converges to a global minimum. Then, we compare the performance of the optimum design predicted using translator B and FEA. Finally, the effects of changing constraints associated with the most important design variables on the objective function of the final optimum designs are studied. Chapter 7 demonstrates the methodology for developing translator B on the A-pillar joint. The objective function and constraints are explained. The translator B is validated and parametric studies are performed to determine the effect of constraints on the optimum designs. Chapter 8 summarizes the work and the conclusions. It also presents recommendations for future work. 22 B-pillar to Roof Rail A-pillar to Roof Rail C pillar to Roof Rail A-pillar to Front Hinge Pillar Shotgun to Front Hinge Pillar Rocker to Rear Quarter Front Hinge Pillar to Rocker B-pillar to Rocker Figure. 1.1: Body Structural Joints 23 Torsion Stiffness I/O Stiffness F/A Stiffness B Front Outside C Fixed A Fixed Figure 1.2: Definition of Stiffness for B-pillar to Rocker Joint 24 Fixed Header Top Roof Rail Front Fixed I/O Stiffness FA Stiffness Torsion Stiffness Figure 1.3: Definition of Stiffness for A-pillar to Roof Rail Joint 25 inputs ( joint dimensions) w11,1 first hidden layer b11 w21,1 w11,2 b12 second hidden layer output layer ( stiffness or mass) b21 b22 Figure 1.4 :Architecture of a Typical Multi-layer Neural Network 26 Structural analysis Neural network simulation analysis Time Training break-even point Analyses for training pairs Number of Analysis Figure. 1.5: Comparison Between Structural Analysis and Neural Network Simulation 27 B-pillar Front Outside Rocker Figure 1.6: B-pillar to Rocker Joint 28 Header Top Roof Rail Front A-pillar Figure 1.7: A-pillar to Roof Rail Joint 29 Chapter 2 Methodology for Developing Translator A 2.1 Introduction Structural analysis programs used to solve design problems are often computationally expensive. Obtaining optimal solutions typically requires numerous iterations involving analysis and optimization programs. This process is often prohibitive due to the amount of computer time required for converging to an optimum design. One promising technique is to simulate a slow, expensive structural analysis program with a fast inexpensive polynomial or neural network. Such a polynomial or neural network is called Translator A in this study. Using polynomials to create regression models is a mature technique. There are a lot of books covering this topic. The methods used in this study are briefly explained in the following sections. The area of neural network is relatively new. A lot of work has been done on the application of neural networks on structural mechanics problems (see Chapter 1 for a review of such applications). A neural network provides a rapid mapping of a given input into the desired output quantities, thereby enhancing the efficiency of the design process. Besides, degeneration of a few computing elements due to hardware problems or due to a few corrupt data, does not seriously affect such networks. Even though significant progress has been made on the theory and application of neural networks in the past few years, there are still many unresolved issues. Such issues 30 include determination of optimal training size to meet the required error tolerance, and determination of the number of neurons in each layer of a multi-layer network. This study develops tools (translators A) using both response surface polynomials and artificial neural networks that can estimate the performance characteristics (stiffness and mass) of a joint once the values of the design variables (dimensions) of the joint are specified. The translator can predict the stiffness and the mass of a given joint almost instantaneously, which will allow for an efficient optimization of the joint or the entire car body. The development of translator A can be divided into three steps. Figure 2.1 shows these steps. First, a parametric model of a structure needs to be created. This model can be easily modified by changing the parameters that specify the dimensions and the topology of the model. Next, a database is created, which includes many feasible designs and their corresponding finite element analysis (FEA) results (stiffness and mass). Polynomial and neural network translators can be developed using the database. Section 2.2 briefly describes the parametric model that is used to create translator A. Section 2.3 explains a procedure for using the parametric model to develop a database for fitting and testing. Section 2.4 explains how to develop and validate polynomial translators using the database. Section 2.5 describes the procedure of developing and validating neural network translators. 2.2 Parametric Models To develop translator A, a parametric model is first created for each joint. CAD packages such as CATIA, I-Deas, and Pro/Engineer can be used for this purpose. The parametric model is generic, and can be easily modified by changing values of the parameters that determine the dimensions and the type of construction. 31 The input variables of the parametric model can be divided into several types. The first type are binary variables, which can take the values of "Yes" or "No". These variables are called control parameters, and they specify the type of construction of a joint. For example, if a branch has a bulkhead then we use a binary parameter to specify that there is a bulkhead in a particular design. The second type can be further divided into two sub types: design variables controlling the shape of the joint, such as the length and height of a branch; and design variables storing the FEA information, such as the thickness of each part. The development of a parametric model consists of four steps. Murphy (1995) explained these steps in detail. Here each step is briefly described. • Identify construction types of the joints by examining actual joint hardware, simplifying the geometry, and classifying joints into different types on the basis of their type of construction. • Parameterize a joint using six level of parameters. The higher levels define the overall joint shape while the lower levels define the details of joint geometry. The design variables of each level are used to define the following: Level 1: Number and position of branches Level 2: Cross sections of branches Level 3: Blending areas Level 4: Internal reinforcements Level 5: Connections Level 6: Openings • Establish constraints and relations between design variables to ensure that a joint design is feasible. The constraints include manufacturing, packaging, styling, and mathematical constraints. • Create a parametric model that can account for different construction types. This model should include all the constraints mentioned in the previous step. It should warn the user when a design violates a constraint. 32 To develop translator A, the type of construction of the joint is fixed. This means that the control parameters are fixed. Only the second type of design variables, which control the shape and FEA characteristics of the joint, can vary. 2.3 Database for Developing Translator A To create and validate a response surface polynomial or a neural network, a database is needed that contains many designs (between fifty to few thousands) and their performance characteristics. These designs, called examples, are used to train a neural network or develop a polynomial. These designs should be selected in a way that they provide the maximum amount of information about the relation between the design variables and the performance characteristics of a joint. This allows us to determine the polynomial or the neural network with good accuracy. Many methods can be used to select the values of the design variables of the joints in the database. A lot of work has been done to compare the advantages and disadvantages of different methods. Design of experiments (DOE) (Box et al. 1960, 1978) is a branch of statistics that provides the researcher with methods for selecting the independent variable values for which a limited number of experiments will be conducted. Experimental design methods create certain combinations of experiments in which the independent variables are prescribed at specific values or levels. The results of these planned experiments are used to investigate the sensitivity of some dependent quantity (response) to the independent variables. Rogers (1994) compared the results of four different methods, namely, hypercube, linear, PROSSS, and random. The hypercube method builds a hypercube around the midpoint of the design space, which corresponds to the initial or nominal values of the design variables. The best choice of points for the hypercube is points at each of the n corners ( 2 V points for nV design variables), the midpoint for each face ( 2 ⋅ nV points), and the midpoint of the design space (the initial design). However, if the corner points are chosen, the number of required fitting pairs quickly becomes very large as nV 33 increases. Therefore, it might be better to try to represent the design space by choosing only the midpoints of the faces. The linear method creates the training pairs starting at the lower bound of each design variable and stepping through the design space until reaching the upper bound. PROSSS (Programming System for Structural Synthesis) is a software that is designed to integrate the Engineering Analysis Language (EAL) structural analysis program and CONMIN optimization program. In PROSSS method, the user first executes the optimization program CONMIN for a few cycles to optimize the structure in which the user is interested (CONMIN does not reach the optimum). A new design is analyzed in each iteration using EAL. PROSSS uses those intermediate designs and their responses obtained from FEA to train the neural network. In random method, designs are created using a random number generator with no values allowed outside the lower and upper bounds of the design variables. Taguchi method is a design of experiments method that is very popular in the industry. In the Taguchi method, a set of orthogonal matrices has already been prepared. The different values in those matrices correspond to the different levels of the design variables. For example, in a two-level matrix, the numbers 1 and -1 in each position of the matrix can denote the two levels of each design variable such as high or low, loose or tight, and good or poor. Taguchi method uses simple charts to describe how to design an experiment, and how to extract the information needed from the experimental results. The user just needs to follow a procedure in a cookbook without knowing statistics theory and details behind this method. For a model with nV variables, there are 2 nV different combinations if each variable only has two levels, and 3nV different combinations if each variable has three levels. In Taguchi method, only a fraction of all possible combinations are used in designing the experiment. For example, consider that there are 3 two-level variables, and we only want to get the effect of each variable on the output, and that we are not interested in the effects of the interactions of different variables on the output. Only four experiments are needed instead of nine if one wants to consider all combinations of values of the variables. The 34 total number of experiments required will increase very quickly if the user needs the effects of the interactions of different variables on the output. However, the method is still more efficient than using all possible combinations of variables. The hypercube, the Taguchi method or other design of experiments methods require the stiffness and mass of joints for many predetermined combinations of values of the joint dimensions that correspond to infeasible designs. Many of these designs violate manufacturing, styling, and packaging constraints, while others have unreasonable geometry and cannot be modeled using a parametric model. Based on these considerations, the random method is used to create designs. This method uses a random number generator to create values of design variables corresponding to joint designs. Each joint design is checked against the constraints and those designs violating the constraints are rejected. The remaining designs are analyzed using a finite element program, such as MSC/NASTRAN to estimate their performance characteristics. These estimates are considered as actual values of the performance characteristics when developing translator A. Several thousand designs are typically checked. The dimensions of all the feasible designs and their performance characteristics are stored in the database that will be used later to create the polynomial and neural network translators. Creating the database is one of the most time-consuming parts of developing translator A. Generally, it takes 2-4 minutes to check if a design is feasible. If a design is feasible, it takes 2-3 more minutes to create the FEA mesh and a NASTRAN bulk data file. If a design causes the parametric model to crash, it takes 1-2 minutes to recover the model. It takes 4-5 minutes to analyze a design using a finite element program like MSC/NASTRAN to get its performance characteristics. For a typical joint with 40-50 design variables, the database contains 500-1000 designs. The author estimates that it takes two to three months to develop such a database. 35 2.4 Developing and Validating a Response Surface Polynomial The degree of a polynomial depends on the relation between the design variables and the performance characteristics. This study uses linear polynomials, second degree polynomials and double regression models, to create the translators. Double regression, first approximates the performance characteristics as a function of the design variables using a polynomial and then uses another polynomial of the estimated values of the performance characteristics to obtain better estimates of these quantities. Stepwise regression is used throughout this study to obtain the best regression model. The best model is the model that fits the data well and contains only important terms, that is terms that affect significantly the response. This section describes how to develop a response surface polynomial. Specifically, it will describe the following tasks: • • • Identification of important design variables. Construction of regression models. Comparison of different regression models, and validation of regression models. 2.4.1 Determining the Most Important Design Variables Generally, many design variables are needed to completely describe the geometry of a joint. For example, the B-pillar to rocker joint considered in this study has 52 design variables, while the A-pillar to roof rail joint has 48 design variables. Among all these design variables, only a few significantly affect the performance characteristics (stiffness and mass) of the joint. Therefore, to develop an efficient polynomial and neural network that simulates the relations between the design variables (input) and the stiffness or mass (output), it is important to determine those design variables that affect significantly the stiffness and mass. One can rank design variables in terms of importance by comparing all possible polynomials with all possible combinations of design variables. This approach is obviously impractical in most real life design problems with many design variables (more than 30). It is also impractical to use stepwise regression on a complete second degree polynomial. For example, a complete second degree polynomial with 50 design variables 36 has 1326 terms. Since stepwise regression considers many combinations of these terms in each step, it is impractical for such polynomials. To overcome these difficulties, we use stepwise regression on a linear polynomial to identify and rank the most important design variables as described below. First, we find the design variable that has the biggest effect on error between the predicted results and FEA results and include this design variable in the model. For this purpose we compare single-variable models, each containing one design variable, where nV (= p-1 for a linear polynomial) is the number of design variables. Then we find the next most important variable among the remaining design variables that when included in the model would result in the biggest reduction in the error and add it in the regression. This process is repeated until all important variables enter in the model. The improvement in the overall fit resulting from adding a design variable is measured by the corresponding significance probability. This is the probability that the actual effect of a term on the fitting error is zero and consequently the observed improvement in the overall fit is due to luck. In every step, a hypothesis test is performed to decide when to stop. Specifically, we test the hypothesis that the actual effect of a parameter on the overall fit is zero. If the reduction in the overall error is statistically significant, the parameter is added to the model. Otherwise, we conclude that there are no important parameters left and the procedure stops. Theoretically, the obtained regression model is the best linear model, and it contains the most important design variables. In every step of linear stepwise regression, the F-ratio of each design variable is calculated. Section 2.4.3 gives the definition of F-ratio. F-ratio can be used as a measure of the importance of design variables: the variable with bigger F-ratio has bigger effect on the output. According to the F-ratios, the design variables are listed from the one with the biggest F-ratio to the one with the smallest F-ratio. This list is used to compare the importance of the design variables, and will be used throughout the following sections. 2.4.2 Choosing a Polynomial Regression Model Before creating a response surface polynomial, we select its degree. Linear polynomials can only simulate linear relations between the independent and dependent variables. Higher degree polynomials are more flexible in simulating complicated input- 37 output relations. On the other hand, higher degree polynomials can have a large number of parameters. If we do not have enough designs for fitting, then the generalization performance will be poor. The reason is that some coefficients cannot be predicted with high confidence using a reasonable number of examples. The total number of parameters in the regression model will increase exponentially as we increase the degree of the polynomial. Moreover, a higher degree polynomial may become ill-conditioned. Another drawback of using a high degree polynomial is that we will have less confidence in the parameters of the polynomial when more parameters are used. Because of these reasons, generally, we use polynomials with degree higher than three in a few cases only. In this study, linear, second degree polynomial models, and double regression models are considered. Linear Polynomials In general, the most parsimonious regression polynomial that fits the data well is the best. Therefore, a linear polynomial should be used if the relation between the independent and the dependent variables is almost linear. This is the case in many real life design problems because there are many constraints that do not allow the design variables to vary outside a narrow zone (Hughes, 1983). Second Degree Polynomials Linear polynomial models can only simulate linear relations between the independent and dependent variables. The fitting and testing results from the linear regression models are not always accurate. Therefore, we consider second degree polynomials for developing translators for some performance characteristics. The most straightforward way of using second degree polynomials is to consider all design variables, all the interactions, and the square terms of all design variables in 0 1 2 stepwise regression. Such a polynomial includes C n + 2C n + C n (here V V V nV (nV − 1)(nV − j + 1) ) terms, which means that we need at least j ( j − 1)1 0 1 2 C n + 2C n + C n designs to determine the coefficients of the polynomial. For the B-pillar C nj = V V V V 38 to rocker joint, the number of design variables, nV , is 50. It is impractical to use a complete second degree polynomial because such a polynomial requires at least 1326 designs to determine the coefficients. To address this problem, we rely on the ranking of the design variables obtained using a linear polynomial. Figure 2.2 explains this process. First, we consider only a small group of the most important design variables found using a linear polynomial. This small group can have about five design variables. We create a complete second degree polynomial using these few most important design variables, and use stepwise regression to determine which terms should be included in the regression model. The stepwise regression yields an incomplete second degree polynomial in which only the important terms are left. Then, we add the next most important design variable among the remaining design variables and its square to the model according to the ranking determined by the linear polynomial. The new regression model considers the new design variable, the square of the new design variable, the remaining terms from the former regression, and the interactions between the new design variable and former design variables. Using stepwise regression, the best regression model can be found again. This process is repeated by adding one variable at a time to the model. At each step, the accuracy of the predictions of the polynomial increases, but the rate of improvement reduces as we add more parameters. We stop when we reach the point of diminishing returns where the improvement in the accuracy becomes insignificant as we add more variables. Double Regression Models When creating the polynomial translators for some joints, it is found that both linear and second degree polynomials are inaccurate. The performance characteristics can be more accurately predicted if we use a second regression to predict the performance characteristics as a function of the predictions of a linear regression model. We can identify this case by observing a scatter plot of the exact values of the dependent variables (performance characteristics of the joint) versus the predictions from the linear regression model. For example, as Figure 2.3 shows, the predictions from a linear regression model are not evenly distributed along the two sides of a straight line inclined at 45 degrees 39 relative to the horizontal axis. Instead, the predictions are almost evenly distributed along the two sides of a cubic/quadratic polynomial of which the predicted value of the performance characteristics are independent variables. Based on this observation, a second regression is performed that uses a cubic/quadratic polynomial to simulate the relation between the predicted results from the linear regression model and the actual performance characteristics. The following is the formulation for the second regression (Fig.2.3). ˆ ˆ y D = ∑ ci ( y L ) i i =0 nD (2.1) ˆ ˆ where y D and y L are the values of the dependent variables predicted from the double regression and linear regression models, respectively. n D is the degree of the double regression model. Coefficients ci in above formulation can be determined using regression. Note that, in some cases, double regression significantly improves the prediction when compared with the linear regression model and the second degree models. 2.4.3 Criteria for Selecting Polynomial Translators When creating the polynomial translators, only a subset of all design variables is considered in the regression model. It is important to know which design variables and what interactions between design variables should be considered in the regression model. This study uses four criteria to determine whether a regression model is appropriate in terms of its generalization performance. • • • • Significance probability Determination factor, R2 Cp criterion AIC (Akaike’s Information Criterion) We describe each criterion in the following. 40 Significance Probability Significance probability is the probability that the actual effect of a term on the fitting error (error between the predicted values of some performance characteristic and the calculated values of this performance characteristic from FEA) is zero. Consider that we want to decide if we should add a particular term to a given polynomial model that contains p-1 terms plus the constant term. The significance probability associated with that term is calculated by finding the value of the F-ratio: F= SSRim − SSRr SSEim /(n f − p ) (2.2) where n f is the number of designs used in fitting the polynomial. Subscript im denotes the model with the additional term (improved model), and r denotes the original model without the term (reduced model). The quantity, n f -p, is called number of degrees of freedom of the improved model. SSR is the regression sum of squares: ˆ SSR = ∑ ( yi − y ) 2 i =1 nf (2.3) ˆ where, yi is the response of the ith design predicted by the polynomial and y is the average of the responses of the n f designs. polynomial: ˆ SSE = ∑ ( yi − yi ) 2 i =1 nf SSE is the sum of square errors of a (2.4) where yi is the measured response of the ith design (in this study, yi is the response calculated using FEA). The F-ratio can be used to test the hypothesis that the additional term is actually zero (this is called null hypothesis). If this were the case, then F would follow the F1,n − p f probability distribution, which has been tabulated in many books (e.g., Draper and Smith, 41 1981). A large observed value of F (say 10) indicates that the model will be significantly improved when the pth term is added to the model. This is evidence that this hypothesis is wrong and consequently that the additional term is non zero. Indeed, if the null hypothesis were right, then it would be very unlikely for F to assume such a high value. To determine if the additional term is important we calculate the significance probability, which is the probability of a variable, which follows the F1,n − p distribution, exceeding the f observed value of the F-ratio (Eq. 2.2). A low probability (say 5%) shows that the additional parameter should be non zero. The significance probability is used to determine how important a term should be in order to be included in the regression model. A significance probability that is too high will result in including terms that are not statistically important, and will reduce the generalization performance of the polynomial. On the other hand, using a significance probability that is too low will not allow us to include all the important terms in the regression model. In most applications, a significance probability between 1% and 10% is used. In our work, we use a 5% significance probability. The determination factor, R2 R2 expresses the percent of the variation in the response of the designs in the database from the mean response that is explained by the model: R2 = SSR SSTot − SSE = SSTot SSTot (2.5) where SSTot is the total sum of squares of the variations of the measurements from the mean. SSE is the sum of square errors defined in Eq. 2.4. A model with a value of R2 close to one is accurate because it explains the biggest portion of the variation in the response of all the designs used for fitting. When using R2, we only include in the model those terms that increase significantly R2. Those terms that result in small increases in R2 are not included in the model because 42 they will corrupt the generalization performance even though they can slightly improve the fitting results. Cp criterion Cp criterion is an alternative statistic to determine the predictive capability of a polynomial. It is calculated using the following equations: Cp = where s2 = ˆ ∑( y j =0 nf j SSE − (n f − 2 p) s2 (2.6a) − y)2 (2.6b) nf − p s2 is an estimate of the variance of the error in the measurements, which is also called surface-fitting sample variance. If we use data from FEA, obviously, there is no measurement error. However, we can still use Eq. 2.6b and calculate s2 from Eq. 2.6b. Cp is compared to the number of terms, p, of a polynomial to find if the polynomial includes all the important terms and only these terms. If Cp is considerably larger than p, this indicates either that the polynomial does not include all important terms or that it includes redundant terms. If we start with a polynomial with very few terms, then Cp decreases as we add important terms. The reason is that this increases the model accuracy and reduces SSE significantly. However, as we continue adding terms we reach a point where redundant terms enter the model. This means that the function representing Cp in terms of p will flatten out or will start increasing (Eq. 2.6a). We should stop adding terms when the value of Cp first approaches the number of terms in the polynomial. Akaike’s Information Criterion, AIC AIC is defined by the equation below: AIC = n f Log ( SSE / n f ) + 2 p (2.7) 43 where n f is the total number of fitting pairs, and p is the number of parameters in the polynomial, including the constant. AIC is similar to the definition of SSE (Sum of Squared Errors) except that AIC also takes into account the effect of p. The model with the smallest AIC value is the best. After creating a polynomial translator by using stepwise regression, the standard deviation is used to test the generalization performance of the obtained translator. Standard Deviation Standard Deviation is used to compare the accuracy of different models. It is defined as nf σ = (∑ (ri − r ) 2 ) (n f − 1) i =1 (2.8a) where ˆ ri = yi / yi 1 r= nf ∑ ri i =1 nf (2.8b) ˆ y i and yi are measured and predicted values of the response, respectively. ri is the ratio of the predicted value over the measured value. An unbiased model has r ≈ 1. A precise model has a standard deviation close to zero because ri ≈ r for every design in the training or testing set. 2.4.4 Validation of Polynomials After obtaining a polynomial, it is important to assess its accuracy when using it to predict the performance characteristics of new designs. As explained in the former sections, a model may fit the data accurately but have poor generalization performance mainly because it contains unimportant terms. Using stepwise regression and the above 44 criteria usually ensures that a polynomial has good generalization performance. This allows using the entire database of examples for fitting. To be on the safe side, one may want to use a portion of the database for validation of a polynomial model. To test the generalization performance of a polynomial. This study splits the database into two groups. One group is used for fitting. Once a polynomial is obtained, we use the other group to test its performance. The standard deviation of the predicted over the actual values is calculated. The polynomial with the best generalization performance should have the smallest standard deviation for the testing results. 2.5 Developing and Validating a Neural Network Neural networks have been studied for many years in the hope of achieving humanlike performance (Lippmann, 1987). Neural networks consist of many linear and nonlinear computational elements called neurons. Neurons are connected via weights and bias that are adapted during the training process to improve the generalization performance of the neural network (Fig. 2.4). Neural networks can be used to simulate complex mapping relations (Hajela and Berke, 1990). In selecting a neural network, we have to select the architecture of the neural network (for example, the number of hidden layers and the number of neurons in each hidden layer), the transfer functions of the neurons, and the training algorithm for determining the parameters of the neural network so that it can predict the response of a given structure (Lippmann, 1987, Alon et al., 1991). The multi-layer perceptron is the most widely used static network model. The back-propagation training algorithm is selected because it is the most popular one and has been implemented in many commercial software packages. Figure 2.4 shows a typical multi-layer neural network. Most applications use neural networks with one or two hidden layers to approximate input-output relations. Cybenko (1989) showed that, a neural network with one hidden layer can approximate any continuous function. However, a neural network with two hidden layers may require far 45 less neurons than a single hidden layer neural network in some cases. On the other hand, a neural network with two hidden layers may need much more training time. In this study, we found that a neural network with one hidden layer can predict the performance characteristics of a joint with acceptable accuracy. 2.5.1 Transfer Function The transfer functions of the hidden layer neurons and the output layer neurons are sigmoid and linear functions, respectively. One advantage of the sigmoid function is that it is differentiable. This property is important because it makes possible to derive a gradient search learning algorithm for networks with multiple layers. Another reason that the sigmoid function is popular is that, it varies monotonically from 0 to 1 as y varies from -ˆ to +ˆ. It is particularly suitable for pattern recognition problems. Using linear function in the output layer makes it easier to change the output in a big range. The sigmoid function can be expressed as: f ik ( x ) = 1+ e 1 − zik / θ (2.9a) where zik is the input to the ith neuron: = ∑ wik j x k − 1 + bik , j j =1 k nn ,i zik (2.9b) Coefficient θ is used to control the steepness of the sigmoid function. The smaller the θ, the steeper the sigmoid function. θ is set to be one in this study. k denotes kth layer. k Subscript i stands for ith neuron in a layer. nn ,i is the number of inputs for the ith neuron in the kth layer. j denotes the jth neuron of the former layer. wik, j is the weight between the ith neuron of kth layer and the jth neuron of former layer. bik is the bias of the ith neuron in the kth layer. x k −1 is the output from the jth neuron of the (k-1)th layer (Fig. 2.4). j 46 The linear transfer function has the following form: f i ( x) = ∑ wik, j x k −1 + bik j k j =1 k nn , i (2.10) 2.5.2 Choosing Inputs and Outputs of a Neural Network To use a neural network to simulate the mapping between the design variables and the performance characteristics of a design, such as an automotive joint, first, we need to decide how many design variables to consider in the neural network. As explained in Section 2.4.1, among all the design variables that are used to describe the geometry of a joint, only a few affect significantly the performance characteristics. Therefore, it is important to consider only those design variables that affect significantly the performance characteristics. First, the design variables are ranked in terms of importance using the results of stepwise regression with linear polynomials (see Section 2.4.1). According to the importance of design variables, first, we only consider a small group of few most important design variables in a neural network. This group can have five or six design variables. Then, we gradually increase the number of design variables in the neural network by adding one variable at a time. We terminate this process when the error of prediction of the testing results falls below a given value. We can use one neural network to predict all performance characteristics of a joint and mass, or one neural network for each quantity. Since the important design variables for one performance characteristic are different from those for another characteristic, the first approach would result in one network with too many design variables. Some design variables of this network may affect only one output, but not the other outputs. On the other hand, the second approach only considers the important design variables for each performance characteristic and the mass. These networks can be more robust than a big network obtained by the first approach because they only contain the important design 47 variables. The drawback of the second approach is that it works with more neural networks than the first approach. This study follows the second approach because translator A must be robust. 2.5.3 Normalization of the Input-Output Sets Some design variables, such as thickness, can only change less than 1 mm, while some other design variables, such as the height of a beam can change in a big range. The magnitude of most input design variables varies from 1 to 10 2 , while the magnitude of the output varies from 10 7 to 108 for stiffness. The magnitude of mass is 10. It is useful to normalize both the input and the output variables so that they can all vary in the same range [-1,1]. The reason is that by normalizing all variables into the same range we can avoid the problem that the output of the neural network is over sensitive to some variables and insensitive to others. Moreover, if the ranges of design variables and the performance characteristics are too big, this can cause numerical difficulties. Li and Parsons (1995) also observed that selecting an appropriate range in the magnitude could reduce the training time by an order of about three. There are many methods to normalize the variables. Two methods are considered and tested in this study. In the first method, the parameters are normalized by dividing each variable by its mean value. The second method normalizes the parameters so that they can vary in the range [-1,1]. It was observed that the normalization method can have an appreciable effect on the final results. Generally, when using the first normalization method, the testing results converge much slower than when using the second normalization method (Fig.2.5). Figure 2.5 shows that the neural network reaches the minimum at about 1300 epoches when using the second normalization method. In neural networks, “epoch” refers to one iteration of the training process. On the other hand, it takes more than 5000 epoches to reach the minimum if the first normalization method is used. The reason is that the transfer function is the sigmoid function (Eq. 2.9), which varies between 0 and 1 when zik changes from -∞ to +∞. [-1,1] is the activating range for the sigmoid function. The convergence speed is related to the derivative of the function f i k (x) with respect to the inputs of the neurons, zik . This derivative is bigger for the 48 same change of zik when zik is within the range [-1,1] than that when zik is outside this range. Since in the second method, all input variables (design parameters of a joint) are normalized so that they can vary in the range [-1,1], the second method generally converges faster than the first. 2.5.4 Determining the Number of Neurons in the Hidden Layer As mentioned earlier, this study uses a neural network with one input layer, one output layer, and one hidden layer. For our problem, the number of neurons in the input layer is equal to the number of design variables considered in the neural network. The number of neurons in the output layer is one for each neural network used to predict each performance characteristic. Only the number of neurons in the hidden can be changed. The number of neurons in the hidden layer affects the generalization performance of the neural network. If there are too few neurons, the neural network will neither learn nor be able to generalize relationships from the training data. On the other hand, if there are too many neurons relative to the number of examples, then the neural network will memorize the training data but will not be able to generalize relationships. The number of unknowns in a neural network is a key factor in choosing the number of neurons in a neural network. For example, a neural network with three layers, including the input and output layers, has ( n1 + n3 + 1)n2 + n3 unknowns, including the weights and bias, where n1, n2, and n3 are the number of neurons in each layer. There are several methods to decide how many neurons to consider in the hidden layer. Some publications use a neural network that has more unknowns than the inputoutput pairs. Most papers select a neural network in a way that the total number of unknowns is less than the total number of the input-output pairs used in training (Carpenter and Barthelemy, 1992). The networks in these papers are determined. Carpenter and Barthelemy (1992) suggested that the number of neurons in the hidden layer should be between the sum and the average of the number of neurons in input and output layers. Hush et al. (1993) suggested starting with the smallest possible network 49 and gradually increasing the number of neurons until the performance of the network begins to level off if one has little or no prior knowledge. Only determined neural networks are used in this study. That is, the number of hidden layer neurons is such that the total number of unknowns is less than the total number of designs for training. For relatively simply mapping relations, the number of design variables is changed from one to the maximum value for which the neural network is determined. The number of neurons in the hidden layer is chosen so that the total number of unknown neural network parameters is approximately equal to 80-90% of the total number of designs in the training set. For complex mapping relations, both the number of design variables and the number of neurons are changed from one to a maximum value to determine the best combination. The neural network with smallest testing error is selected as the translator. 2.5.5 Method of Determining When to Stop Training First, the network is trained to implement a mapping that matches the examples in the training set as closely as possible. Back-propagation, which is an iterative process , is used. There are several stopping criteria. The first is based on the magnitude of the gradient of the total error in training with respect to the parameters (weights and biases). The iteration process is terminated when the magnitude of the gradient is sufficiently small, since by definition, the gradient will be zero at the minimum. Second, one might stop the training when the sum of square errors, SSE, falls below a fixed value. However this requires to know the minimal value of SSE, which is not always available. Third, one might stop the training process when a fixed number of iterations has been performed. However, there is little guarantee that this stopping condition will terminate the training when the sum of square errors of the testing results reaches its minimum. Forth, the method of cross-validation can be used to monitor the generalization performance during training, and terminate training when there is no longer an 50 improvement. Cross-validation splits the data into two sets: a training set, which is used to train the network, and a test set which is used to measure the generalization performance of the neural network. During training, the performance of the neural network on the training data will continue to improve, but its performance on the test data will only improve to a point, beyond which it will start to degrade. The training process is terminated at this point, because the network starts to overfit the training data. Crossvalidation is explained in Fig. 2.6. The solid and the dashed lines denote the standard deviations of the training and testing results versus the number of epoches, respectively. It is observed that the standard deviation of the training results consistently decreases when increasing the training time. However, the testing results only improve to a point (at about 1300 epoches). Beyond this point, the standard deviation of the testing results begins to increase. Amari, Murata, and Muller (1995) discussed the problem of overtraining and crossvalidation. They demonstrated that when n f > 30 nm , where nm is the size of neural network (number of weights and biases), and n f is the number of training pairs, crossvalidated early stopping does not improve the generalization error. Indeed, no overtraining was observed on the average in this range. For most practical problems, this range of n f is often inaccessible due to the limited size of the data pairs. The paper also demonstrated that early stopping does improve the generalization ability to a large extent for n f <30 nm . This paper proposed that, for large nm , only 1 / 2n m percent of inputoutput pairs should be used to determine the point of early stopping in order to obtain the best performance. The first three criteria are sensitive to the parameters of the solution process, such as the minimum value of the total error, which are difficult to determine. Cross-validation, however, does not have this problem. This study uses cross-validation. The available designs are split into three sets: a training set, and two testing sets. The first testing set is used to determine when to terminate the training process, while the second testing set is used to test the generalization performance of the neural network after training has been completed. 51 An additional benefit of cross-validation is that the resulting neural networks are insensitive to the number of neurons in the hidden layer (Chapter 4). 2.5.6 Method of Testing the Generalization Performance of the Trained Neural Network After training a neural network, we test its generalization performance by using it to predict the performance of designs that are not used in training. The second testing set is used to test the trained neural network. After getting the testing results, the ratio of the predicted result over the FEA result can be calculated. The standard deviation of this ratio (Eqs. 2.8a and 2.8b) is used to measure the accuracy of the neural networks and decide if a neural network is acceptable. The same measure is used to compare the results from neural networks with different architectures (number of neurons in hidden layer, and number of input variables). The neural network with the smallest standard deviation for the testing results is chosen as the neural network translator. We also compare neural network with response surface polynomials according to their standard deviations. 52 Create a parametric model Determine the ranges of design variables Generate random designs for a database N Design feasible? Y Put in database N Enough designs ? Y Create polynomial translators Create neural network translators Choose best polynomial and neural network translators Figure 2.1: Method for Creating Translator A 53 Rank of design variables in terms of importance Choose most important one in the remaining design variables Choose the most important 5-6 design variables Create complete second degree polynomial Remaining terms+ new design variable+ interactions + square of new design variable Start with a constant term Choose the term with the biggest F-ratio in the remaining terms Important term ? N Stepwise regression Keep important terms Test the incomplete polynomial N Meet stop criterion? Y Second degree polynomial Y Exclude unimportant terms Figure 2.2: Method for Developing Translator A Using Second Degree Polynomials. 54 Relation Between Predictions from Linear Regression Model and FEA Resuls and Explanation of Double Regression 1.8E+08 1.6E+08 1.4E+08 1.2E+08 FEA Results 1E+08 8E+07 6E+07 4E+07 2E+07 0 0 2E+07 4E+07 6E+07 8E+07 1E+08 1.2E+08 Predictions from Linear Regression Model Figure 2.3: Comparison of Predictions from Linear Polynomial Model and FEA Results and Explanation of Double Regression 55 inputs ( joint dimensions) w11,1 first hidden layer b11 w21,1 w11,2 b12 second hidden layer output layer ( stiffness or mass) b21 b22 Figure 2.4: Architecture of a Typical Multi-layer Neural Network 56 Comparison of Two Different Normalization Methods 0.40 0.35 Standard Deivation of Testing 0.30 Normalized to [-1,+1] Divided by mean 0.25 0.20 0.15 0.10 0 10 20 30 40 2 50 60 70 80 Time of Training (×10 epochs) Figure 2.5: Comparison of Two Different Normalization Methods 57 Relation Between Standard Deviations and Time of Training 0.16 Training Testing 0.14 Standard Deviation 0.12 0.10 0.08 0.06 0 10 20 2 30 40 Time of Training ( ×10 epochs) Figure 2.6: Relation Between Standard Deviation and Time of Training 58 Chapter 3 Methodology for Developing Translator B 3.1 Introduction Translator B is a tool that finds the most efficient, feasible joint design that meets given performance targets. An efficient joint has the lowest mass and its stiffness is as close as possible to a given target. In this study, we refer the most efficient feasible design as the optimum joint design. This tool is important at the early design stage because it links performance requirements for a joint to the design variables and mass of a joint. As explained in Chapter 1, this tool will allow designers set up meaningful performance targets. Translator B finds the optimum design using numerical optimization methods. The objective function to minimize can be the mass of a joint or some combination of the mass and stiffness. Design variables include both the dimensions of a joint and variables that specify FEA information (e.g., plate thicknesses). Constraints can be classified under packaging, styling, manufacturing, mathematical, and performance target constraints. Section 3.2 describes the formulation of the optimization problem for translator B. Section 3.3 explains and compares two optimization algorithms used in this study. Section 3.4 studies the verification and validation of the optimization results. 59 3.2 Formulation of the Optimization Problem A general optimization problem can be expressed in the following form. Find To minimize Subject to g j ( X ) ≤ 0, j = 1, NCON xiL ≤ xi ≤ xiU , i = 1, NDV (Inequality constraints) (Side constraints) (3.1b) X = { x 1 , x 2 ,  , x NDV } T F(X) (3.1a) where X is the vector of design variables. xi is the ith design variable. xiL and xiU are the lower and upper bounds of ith design variable. NCON and NDV are the number of constraints and the number of design variables, respectively. There is no equality constraint in equation 3.1b. The reason for this is that the optimization program does not directly handle equality constraints. The program always converts an equality constraint into two inequality constraints. For example, the program converts the following equality constraint: h( X ) = 0 (3.1c) into two inequality constraints: h( X ) ≤ 0 h( X ) ≥ 0 (3.1d) Another way of handling equality constraints is finding an expression of a design variable in terms of other design variables from the equality constraint. This method can reduce the number of design variables of the optimization problem. Therefore, it can increase the speed of convergence and improve the reliability of the optimization program. Unfortunately, it is not always possible to find an explicit expression of a design variable from an equality constraint. 60 3.2.1 Selection of Design Variables Design variables are physical design variables that define the geometry and specify FEA information. These design variables are linked to the performance characteristics of the joint through translator A. 50-200 independent design variables are generally needed to completely describe a joint. As explained in Chapter 2, among all the design variables that are needed to completely describe a joint, only a few affect significantly its performance characteristics. Therefore, it is important to consider only these important design variables in optimization to reduce the size of the problem and avoid numerical problems. Solving an optimization problem with even a modest number of design variables can be very difficult if the gradients of the objective function and constraints with respect to design variables are very small or zero. Moreover, only a few design variables affect significantly the objective function. Among all design variables, some are generally fixed by the designer because of manufacturing, packaging, or styling considerations. For example, the length of a branch, the width of flange and the distance between two adjacent spot welds are often fixed because of manufacturing consideration. Member lengths are fixed so that different joints can be fairly compared. Some design variables are not independent for a particular type of construction. For example, the depth of the pillar reinforcement can be determined from other design variables. In translator B, design variables are divided into four types: • • Type 1: design variables that are fixed by the optimization program, such as the width of a flange. Type 2: design variables that are fixed by the user, such as the orientation of a branch. These variables are usually specified from the general shape of the car body and they cannot change in optimization of a joint. • Type 3: dependent design variables whose values can be derived from other design variables. The values of dependent design variables are automatically updated when other variables change. • Type 4: independent design variables that can change in optimization. 61 The number of variables that are fixed by the optimization program and the number of dependent variables are fixed for a particular joint. The user specifies which variables are fixed. The user of translator B can easily change the numbers of independent and fixed variables by moving variables from one group to another. To do so, the user needs to change the input file only. This feature makes the optimization program easier to use and more effective than an optimization program that requires changing the source code to move a variable from one group to another. Consider that the user wants to optimize both plate thicknesses and shape variables. In practice, the thickness of a plate assumes discrete values because of manufacturing considerations. The user can initially optimize both the shape variables and thicknesses by treating them as independent variables. Then, the user can fix each plate thickness at the allowable value that is closest to the optimum. An improved design can be found by only optimizing the shape variables in this step. This feature of translator B has some additional benefits: • The user may not be interested in changing all design variables to get the optimum. Instead, the user can change only these design variables that should affect significantly the mass and stiffness. • For a particular joint, the values of some design variables are derived from styling or other requirements, and are usually fixed. Although the user can not change those design variables in some cases, the user may want to perform a parametric study to determine the effect of releasing these variables on the optimum design. Translator B allows to perform such parametric studies easily. 3.2.2 Objective Function Objective function is the target that the user wants to minimize. In our study, the mass of the joint can be chosen as the objective function. The best design should have the minimum mass among all designs that satisfy stiffness requirements (i.e. larger than the required stiffness) as well as manufacturing, packaging, styling and mathematical constraints. 62 However, in practice, a designer may want to design a joint that not only has low mass but also has performance characteristics (stiffness) close to some performance targets. These targets are determined from the optimization of the overall car body using a concept model or they are derived from experience. To account for this situation, the objective function is more generally defined as the weighed sum of mass and some measure of the difference of between performance targets and the corresponding performance characteristics of a design. The general form of objective function can be expressed as follows: ˆ ˆ ˆ F = αM + (1 − α ) (( K I / O − K I / O ) / K I / O ) 2 + (( K F / A − K F / A ) / K F / A ) 2 + (( K Tor − K Tor ) / K Tor ) 2 (3.2) where α is a weight that varies form zero to one. M is the mass of the joint. K I / O , K F / A , ˆ ˆ ˆ and K are the I/O, F/A and torsion stiffnesses of the joint. K , K , and K are Tor I /O F/A Tor the user-specified requirements (targets) for I/O, F/A and torsion stiffness. K I / O , K F / A , and K Tor are nominal stiffness values. Generally, the mass varies from 2 to 8 kg. Each normalized stiffness term in the radical varies from 0 to 3. We normalize the stiffness so that the mass and the stiffnesses can have compatible magnitude. α is set to be one if we only want to consider mass in the objective function. On the other hand, if it is more important for the stiffnesses to be close to their target values, then one should use a small value of α (i.e., α ≅0.1). Translator A predicts the stiffness and mass of a joint in Eq.3.2. This makes translator B very efficient and robust because translator A can give accurate prediction almost instantaneously, even for the infeasible designs whose stiffness and mass can not be obtained using a parametric CAD model. 63 3.2.3 Constraints In the design of a joint, the interaction between different components of the car, such as the position and shape of doors and seats, and the joint should be taken into account. The joint should be manufacturable and should meet the styling requirements. In general, we use many constraints (a few dozens to more than one hundred) to ensure the feasibility of the optimum design. The constraints can be classified as packaging, manufacturing, styling, mathematical, and performance target constraints. Each type is explained below: • Packaging constraints are related to the arrangement of car components in space. They are used to ensure that the joint is compatible with the adjacent components. For example, the width and the angle of the inner part of the rocker should be constrained within a specified limit to leave enough space for the seat. The dimensions door_edge_height and door_edge_width of the B-pillar to rocker joint should be large enough to accommodate the sealant. • Manufacturing constraints are due to manufacturing limitations. They are used to ensure the parts of a joint can be manufactured from a piece of sheet metal. Manufacturing constraints in this study are classified into stamping and welding constraints. Stamping constraints include constraints for plastic strains, die lock and sprint back. For example, the angle between two adjacent plates can not be too small to avoid large plastic strains. The angle between the upper and lower edges of the rocker section should be constrained to avoid die lock. The depth of draw should not be too large relative to its width. • Styling constraints are due to the styling requirements. For example, in B-pillar joint, the flange at the bottom of rocker section should not be visible to a person standing on the right side of the car. • • Mathematical constraints are used to control the range of different design variables to ensure that a design has a feasible geometry. Performance target constraints are the minimum values of the stiffnesses that the designer wants to achieve through optimization. 64 3.3 Optimization Algorithms used in Translator B Because the objective function and some constraints explained in the former section are nonlinear, the optimization problem involved is a nonlinear constrained optimization problem. We can use several methods, including, Modified Feasible Direction Method (MFD), Sequential Linear Programming (SLP), and Sequential Quadratic Programming (SQP), to solve a nonlinear optimization problem. Two optimization algorithms, modified feasible direction method and sequential linear programming, are used in this study. The MFD is reliable and uses the least computer memory (Vanderplaats Research & Development, Inc, 1995). SLP is often most efficient for general applications in terms of the number of function evaluation required. SLP works well for problems that are almost linear. SQP is considered theoretically best if the optimization problem is “well conditioned”. SQP is good for problems in which the objective function and the constraints can be approximated by quadratic functions. In many real life problems, design variables vary in a very narrow band because there are many constraints (e.g., a few dozens to hundreds). SLP and SQP work well in these problems because in the region of the design variables, the objective function and the constraints are close to a linear or quadratic surface. Only the MFD and SLP are considered in this study. SQP is not used in developing translator B. The reason for this is that because some constraints are linearly related to design variables, it is not efficient to use sequential quadratic programming. It was also found that, when using sequential quadratic programming, the optimization program could not find a feasible design in a few cases when the starting point is in the infeasible domain (even though a starting design satisfies all other constraints, it may not satisfy the stiffness requirements, and thus it is an infeasible design). In the following we describe the MFD and SLP methods: 65 3.3.1 Modified Feasible Direction Method The modified feasible direction (MFD) method is a powerful general method, and can be applied to most constrained nonlinear problems. MFD is a modification of the classical steepest descent method. In the steepest descent method, the search direction is the gradient of objective function and the active and/or violated constraints. During each iteration, the search direction is always perpendicular to the former search direction, which makes the steepest descent method very slow to converge when the design approaches the optimum. The conjugate search direction represents a simple modification to the steepest descent algorithm, but provides a major improvement in efficiency. When using conjugate directions, each search direction uses the steepest descent direction plus some fraction of the previous search direction. This algorithm is extremely simple but it dramatically improves the rate of convergence to the optimum (Vanderplaats Research & Development, Inc, 1995). The modified feasible direction method takes into account not only the gradients of objective function and the retained active and/or violated constraints, but also the search direction in the former iteration. Let X 0 be an initial X vector. We update the design according to the following equation: X q = X q −1 + λ × S q (3.3) where S q is the search direction. λ is a scalar whose value is determined through a onedimensional search. Different optimization methods are characterized by different methods to determine the search direction S q . The search direction S q in MFD is determined using the Fletcher-Reeves conjugate direction method when there is no active or violated constraint. S q = −∇F ( X q−1 ) + βS q−1 (3.4a) 66 where β= | ∇F ( X q −1 ) |2 | ∇F ( X q − 2 ) |2 (3.4b) The method of finding search direction S q becomes complicated when there are active and/or violated constraints (Vanderplaats Research & Development, Inc, 1995, Haftka et al., 1992)). 3.3.2 Sequential Linear Programming The basic concept of sequential linear programming (SLP) is very simple. First, a Taylor series approximation to the objective function and constraints are created. Then, this approximation is used for optimization, instead of the original nonlinear functions. This makes it easier to get the values of the objective function and constraints during the optimization process. Also, since the approximate problem is linear, the gradients of the objective and constraints are available directly from the Taylor series expansion. The Taylor series expansion has the following form: F ( X ) ≅ F ( X q −1 ) + ∇F ( X q −1 )T δX g j ( X ) ≅ g j ( X q −1 ) + ∇g j ( X q −1 ) T δX where j∈J (3.5a) (3.5b) (3.5c) δX = X q − X q−1 and J is the set of active and/or violated constraints. Note that everything in the above equations is constant except for the values of X q . These equations can be rewritten as: F ( X ) ≅ F 0 + ∇F ( X q −1 ) T X g j ( X ) ≅ g 0 + ∇g j ( X q −1 ) T X j where j∈J (3.6a) (3.6b) 67 F 0 = F ( X q−1 ) − ∇F ( X q−1 )T X q−1 g 0 = g j ( X q−1 ) − ∇g j ( X q−1 )T X q−1 j j∈J (3.6c) (3.6d) Now the approximate optimization problem becomes: Minimize Subject to: g j ( X q ) ≅ g 0 + ∇g j ( X q−1 ) T X ≤ 0 j xiL ≤ xi ≤ xiU where xiL = xiq − ϑ | xiq | xiU = xiq + ϑ | xiq | (3.7c) i = 1, NDV j∈J (3.7b) F ( X q ) ≅ F 0 + ∇F ( X q −1 ) T X (3.7a) ϑ is a scalar multiplier. Its value is sequentially reduced during optimization. The above linear approximate problem can be solved using different methods such as the simplex method. Some computer programs use MFD to solve this problem because it solves linear problems nicely and works well under most conditions (Vanderplaats Research & Development, Inc, 1995). We recommend using both the MFD and SLP for optimization. If one method consistently provides better results than the other, it can be selected for translator B. 3.4 Validation of Results of Translator B There are several possible problems in optimization, including: • • • • Optimization may not converge. Optimization may converge to a local minimum instead of a global minimum. Optimization may lead to a design that appears to satisfy performance constraints but is actually infeasible because translator A overestimates stiffness. The user may specify performance targets that cannot be achieved. For example, the stiffness requirements can be too high. 68 One reasonable approach is to start from a relatively simple version of the problem that has less design variables and constraints than the original design problem (this can be done by fixing more design variables and removing some constraints). After solving the simplified problem, we can solve more complicated versions of the problem by adding more design variables and constraints. Finally, we can solve the real problem with confidence. Generally, nonlinear constrained optimization problems are computationally difficult to solve and suffer from the fact that the optimization program may converge to a local minimum instead of a global minimum. Several methods can be used to check the results of the optimization program. • One way to check whether the global minimum is found is to solve the same problem from different starting points. If we always get the same solution, then we can have confidence that the global minimum is found. • Another way is to use the optimum design as a starting point, and solve again the optimization problem to see if the optimum design changes. If the second optimization yields the same point consistently, we can conclude that the optimization procedure has converged. • We check if we get the same optimum using different optimization algorithms to solve the same problem. Whenever possible, we should also solve the same problem using different software package. For example, one can use a commercial Fortran optimization program and Mathematica (Maeder, 1997) to solve the same problem. The above checks help ensure that the optimization does converge to the global optimum solution. We also need to check if the optimum design is actually feasible. This is important because in many design problems, the optimizer tends to take advantage of errors in predictive models (translator A in our study) and thus it yields a design that actually has high weight and unacceptable performance (low stiffness). 69 For this purpose, the optimum design is visually checked using a CAD program, such as Pro/Engineer, to see if the optimum design has a reasonable shape. To validate the optimization results of translator B, we select a number of random performance targets (stiffness requirements). Using translator B, we obtain the optimum design corresponding to each set of stiffness requirements. Then, a generic CAD model is used to create the models, and generate FEA data files. These models are analyzed using a FEA program, such as MSC/NASTRAN, to obtain the performance characteristics (stiffnesses and mass) of the optimum designs. By comparing the optimization and FEA results, we can find if there is any general trend between the optimization and FEA results. We also calculate the correlation coefficients between the optimization and FEA results for the stiffness in different directions and the mass of the joint. In general, a correlation coefficient larger than 0.9 shows good agreement between predictions of translator B and FEA results. We also use the stiffness of actual car joints as performance targets and find the optimum design that satisfies these targets. We compare the optimum design from translator B to the actual design. We try to identify and explain trends. We also compare the stiffness of the optimum design predicted by translator B with FEA results. For an optimum design, the values of some design variables are close to the boundary of the region corresponding to the ranges of the design variables in the database. The user may only be interested in changing the design variables in a small range. In such case, the user can narrow the ranges of the design variables. Finally, the relation between the objective function and some important design variables are studied by modifying the lower and upper bounds of these variables and optimizing the joint again. The results should be checked to identify trends and try to explain them. These results should give the designer useful information about improving the design. 70 Chapter 4 Developing Translator A for the B-pillar to Rocker Joint 4.1 Introduction This chapter applies the general methodology for developing translator A to the Bpillar to rocker joint (Fig. 1.1). This chapter is organized as follows: Section 4.2 describes the geometry of the Bpillar to rocker joint, and some common types of construction. Section 4.3 explains how to create a database for developing translator A. Section 4.4 explains in detail the development and validation of translators A for the B-pillar to rocker joint using both response surface polynomials and neural networks. Finally, Section 4.5 discusses the results of translator A for B-pillar to rocker joint. 4.2 Description of the B-pillar to Rocker Joint The B-pillar to rocker joint, which lies between the front and rear doors, is a T- like joint (Figs. 4.1a, 4.1b and 4.2). The vertical branch is called B-pillar, and the horizontal branch is called rocker. The B-pillar supports the rear door. The front door latch is also attached to the B-pillar. The rear door of the car is connected to the B-pillar using hinge. The stiffness of the B-pillar to rocker joint affects significantly the overall stiffness of the passenger cabin of the car. The performance of the B-pillar to rocker joint is characterized by its I/O, F/A and torsion stiffnesses. Figure 4.1b shows the definition of these stiffnesses. Murphy (1995) established a sound, generic method for parameterizing joints. He created a parametric model for the B-pillar to rocker joint that accounted for most 71 important characteristics of the geometry of the joint. However, this model had the following limitations: • Since the direction of the B-pillar was fixed relative to that of the rocker in the original model of B-pillar to rocker joint, the user could not change the orientation of the B-pillar. • Murphy’s model crashed frequently when changing the values of design variables because some methods he used in creating the model were not very robust. For example, he used a fixed length between spot welds to create these spot welds. • In general, Murphy did not put enough constraints to control the ranges of dimensions. This study improved the parametric model. The new procedure for developing the parametric model uses the relative length between spot welds to create the spot welds. The author added constraints on the parametric model of the B-pillar to rocker joint to ensure feasibility of a design, and provide guidance to the user for correcting the model. The author also changed the sequence of creating different entities in the model to make the model more reliable. The generic model is briefly explained in the following sections. The figures with a “*” sign were extracted from Murphy’s thesis. Most B-pillar to rocker joints have at least five parts (Figs. 4.2-4.3b), namely, • • • • • Front part of the B-pillar (Front rocker) Back part of the B-pillar (Pillar back) Center plate of the rocker (Center plate) Back part of the rocker (Back rocker) Pillar reinforcement (Pillar reinforcement) The names in the parentheses are used in the rest of the dissertation and in the computer program for translator A. 72 4.2.1 Types of Reinforcements To increase the stiffness of the B-pillar to rocker joint, different types of reinforcements are generally used in the joint, including, • • • A pillar reinforcement that extends into the rocker and forms a box (Fig. 4.3a). A pillar reinforcement that does not extend into the rocker (Fig. 4.3b). A rocker reinforcement that is parallel to the upper plate of the rocker and covers the hole formed by the connection of the rocker and pillar (called pillar bridge in this study) (Fig. 4.4). • Transverse bulkheads in the rocker (Fig.4.5). From the above we observe that there are two types of pillar reinforcements. In the first type, the reinforcement extends into the rocker, and forms a box inside the rocker (Fig. 4.3a). In the second type, the pillar reinforcement does not extend into the rocker. All hardware joints we examined had this type of reinforcement also had a pillar bridge (Figs. 4.3b and 4.4). Since both types of pillar reinforcements are widely used in the automotive industry, our model can account for both types. Different types of reinforcements are not always compatible. For example, an extended pillar reinforcement is incompatible with a pillar bridge. On the other hand, if the non-extended pillar reinforcement is used, the pillar bridge is generally used. Bulkheads can be used for any joint. 4.2.2 Types of Rocker Cross Section The generic B-pillar model can account for two different types of rocker cross section (Fig.4.6). In the first type (generic type of rocker cross section), the front part is connected to the center plate, and the center plate is connected to the back part of the rocker. In the second type (non generic type of rocker cross section), the front part of the rocker is connected with the back part of the rocker, and the center plate is connected to the front part of the rocker at the bottom. The B-pillar to rocker joint model in this study can account for both types of rocker cross section. Joints with the first type of rocker cross section are easier to manufacture because the front part of the joint is manufactured 73 from the same piece of sheet metal as the side shell of the car body. The joint with the first type of section also has lower mass than the joint with the second type of cross section because the front plate of the rocker is usually thicker than the center plate. We compared the FEA results for three pairs of joints. The joints in each pair were identical except that one had a generic type of cross section, whereas the other had a non-generic one (Table 4.1). It is found that the stiffnesses of the two joints that have the same dimensions but different types of rocker cross section are practically the same. The joint with the generic type of rocker cross section has slightly lower mass than the joint with the non-generic type of section, but the difference is small (less than 5%). 4.2.3 Type of Construction used in Developing Translator A The reinforcement has big effects on the stiffness and mass of the B-pillar to rocker joint. Joints with the extended pillar reinforcement have higher stiffness than joints that have the non-extended pillar reinforcement and pillar bridge (Murphy, 1995). We examined several hardware joints, and listed the dimensions and type of construction of each joint. After consulting with engineers from an automotive company, we chose the following type of construction: There is an extended pillar reinforcement but no rocker reinforcement (pillar bridge) and no bulkhead. The center plate is connected to the front part of rocker and the back part of the rocker (generic type of rocker cross section). Figures 4.7*-4.15 show the dimensions used to define the geometry of B-pillar to rocker joint. Table 4.2 presents the dimensions of the hardware joints examined in this study. 4.3 Developing a Database for Translator A The development of translator A involves two steps. First, we create a database that includes the values of the dimensions for many designs and their corresponding FEA results. The designs in the database are used as examples to teach translator A how to predict the performance of a design. Second, we use response surface polynomials and 74 neural networks to simulate the mapping relations between the dimensions and the performance characteristics of the joint (stiffness and mass). In this study, we first measured the dimensions of the actual joints that were available (Table 4.2). We found the maximum and minimum values of each design variable. As explained in Chapter 2, some design variables are generally fixed because of manufacturing, styling, and packaging requirements, such as, the width of the flange (length_of_flange), the distance between two adjacent spot welds (spot_weld_spacing), and the position of spot welds on the flange (spot_weld_placement). Some variables depend on other variables. By observing the hardware joints, we found that, the values of the blending radii at the outside of the B-pillar are generally close to values of corresponding blending radii at the inner side of the B-pillar. For example, the value of fwd_outer_ver_blending_rad is close to the value of fwd_inner_ver_blending_rad, and aft_outer_hor_blending_rad is close to the value of the dimension aft_inner_hor_blending_rad. For all the joints we examined, the B-pillar branch was always vertical. In other words, the two angles defining the orientation of B-pillar, namely, pillar_io_angle and pillar_angle, were equal to 90 degrees (Fig. 4.7). According to these observations, we assumed that the dimensions related to the flange and spot welds are fixed, the blending radii at the outside of B-pillar are equal to their corresponding blending radii at the inner side of B-pillar, and the two angles defining the orientation of B-pillar are equal to 90 degrees. Table 4.3 shows the range and states (fixed, dependent or free to change) of each design variable. We used a random number generator to create random designs. The Pro/Engineer model was then used to check if a design is feasible. Infeasible designs were screened out. For each feasible design, we updated the parametric model and created a NASTRAN bulk data file. Figure 4.16 shows a FEA model. Then, all the feasible designs were analyzed using MSC/NASTRAN to get their stiffnesses and masses. Among all random designs, only about 20-25% designs were feasible. The dimensions of all the feasible designs and their performance characteristics were stored in a database 75 that was used later to create the polynomial and neural network translators. There are 600 designs in the database for the B-pillar to rocker joint. 4.4 Developing Translator A Two different methods, namely, response surface polynomials and neural networks, were used to develop translator A. To develop translator A, we first ranked the design variables in terms of their effect on the stiffnesses and mass using a linear polynomial (see Chapter 2 for a detailed explanation). Linear and second degree polynomials were used to develop the polynomial translators for the B-pillar to rocker joint. Stepwise regression was used to find the most appropriate regression model (stepwise regression was explained in Chapter 2). We compared the results of the linear and second degree polynomials, from which we chose the best polynomial translators for the stiffness and mass. Neural networks with different number of input design variables and number of hidden layer neurons were studied and compared, from which we chose the best neural network translators. We describe each step in the following sections. 4.4.1 Ranking Important Design Variables Many design variables are needed to completely describe the geometry of a joint. However, only a few affect significantly the stiffness and mass of a joint. A good model for predicting the performance of a joint should include only the important design variables. It is also important to know the relative importance of each design variable because this information can help designers find the most effective way to improve joint design. The design variables were ranked in terms of importance using a linear regression model. Tables 4.4-4.7 rank the important design variables for the I/O, F/A, torsion stiffnesses, and mass, respectively. 76 4.4.2 Developing Polynomial Translators We use both linear and second degree polynomials to develop the translator A for the B-pillar to rocker joint. Each polynomial is presented in detail in the following. Linear Polynomial Models First, we obtained the linear polynomial model using stepwise regression. We developed a linear model that included the important design variables. Table 4.8 presents the results for the linear regression model. The linear polynomials for the I/O, F/A, and torsion stiffnesses had 23, 21, and 23 terms, respectively. The linear polynomial for mass had 31 terms. The standard deviations of the ratio of predictions over FEA results for fitting are 13.5% and 7.2% for I/O stiffness and torsion stiffness, respectively. For F/A stiffness and mass, the standard deviations of the fitting results are 5.1% and 0.6%, respectively. Second Degree Polynomial Models The predictions from the linear regression models were not satisfactory for some performance characteristics. The second degree polynomials were found more accurate than the linear polynomials for the I/O and torsion stiffnesses (Table 4.8). The standard deviations of fitting results for I/O and torsion stiffnesses were reduced to 8.8%, and 6.5%, respectively. The standard deviations of the fitting results for F/A stiffness and mass were 5.2%, and 1.2%, respectively. Figures 4.17-4.20 show the relation between number of design variables in the second degree polynomial and the standard deviations of fitting and testing results for the I/O, F/A, torsion stiffnesses and mass. Section 4.5 discusses the results and compares them with results from neural networks. 77 4.4.3 Developing Neural Networks Neural Networks were used as an alternative to response surface polynomials to simulate the mapping relations between the design variables and the stiffness and mass of the joint. To develop a neural network for simulating the mapping between the design variables and the performance characteristics of a joint, first, we needed to decide how many design variables to consider. We used the rank found using a linear polynomial to determine which design variables should be considered in the neural network. According to the rank of importance of design variables, first, we considered the most important design variable in the neural network. We gradually increased the number of design variables considered in developing the neural network by adding one design variable at a time. Each neural network was trained using a set of training designs (300 designs in this study). Cross-validation was used to determine when to stop training. Then, we tested the trained neural network using a set of new designs that the neural network had not seen before. The neural network that had the smallest standard deviation according to the testing results was chosen as the neural network translator. Since the range of each design variable varies greatly, some design variables, such as thickness, can only change less than 1 mm, while some other variables, such as the overall height of the rocker can change in a big range. The magnitude of some design variables varies from 1 to 10 2 , while the magnitude of the output varies from 107 to 10 8 for stiffness. The magnitude of mass is 10. It is useful to normalize both the input and the output variables so that they can all vary in a smaller range. This makes it more efficient to develop a neural network and improves the robustness of the network. Two methods were tested in this study. The first method normalizes the design variables and the performance characteristics by dividing them by their mean values. In the second method, the parameters were normalized so that they could vary in the range [-1,1]. It was observed from Fig. 4.21 that results from the first normalization method converges much slower than those using the second normalization method. The neural network reached the minimum at about 1300 epoches (an epoch is one iteration during the training of a neural network) when using the second normalization method (solid line 78 in Fig. 4.21). It took more than 5000 epoches for this neural network to reach the minimum if the first normalization method was used. If we train most neural networks over a very long period, we overfit the examples. Although the resulting networks may fit the examples well, their generalization performance may be poor. That is, the network does not predict accurately the stiffness of designs that it has not seen. Figure 4.22 shows the relation between the standard deviation and training time for training and testing results. It is found that the error of the testing result (dashed line in Fig. 4.22) first decreases as training time increases. The error corresponding to testing results reaches the minimum at about 1300 epoches, and then gradually increases when the training time increases even though the error of the training results consistently decreases. To avoid overtraining the network, we used cross-validation. Specifically, we split the 600 designs into three groups: 300 designs were used to do training; 100 designs were used to monitor the training process and to determine when to stop training; and the remaining 200 designs were used to test the generalization performance of the trained neural network. Cross validation can prevent the neural networks from being over trained, and thus improves the generalization performance. The effects of the number of hidden layer neurons on the predictions of neural networks were studied. Figure 4.23 shows the effect of the number of neurons in the hidden layer on standard deviations of the testing results for I/O stiffness when 6, 12, 13, and 24 design variables were considered. Cross-validation was used in obtaining those results. It was observed that the standard deviations of the testing results were insensitive to the number of hidden layer neurons. The difference between the standard deviations corresponding to different number of neurons is probably due to noise. The lack of sensitivity of the accuracy of the neural networks to the number of neurons in the hidden layer is probably due to the use of cross-validation in training. Cross-validation prevented neural networks with large numbers of hidden layer neurons from overfitting the examples. 79 We only used determined neural networks to develop the neural network translator A. A determined neural network involves less unknown parameters than designs used for training. For relatively simple mapping relations, such as the F/A, the torsion stiffness, and mass, we changed the number of design variables from one to the maximum value for which the neural network is determined. The number of neurons in the hidden layer was chosen so that the total number of unknown neural network parameters was approximately equal to 80-90% of the total number of designs in the training set. For complex mapping relations, such as the I/O stiffness, both the number of design variables and the number of neurons were changed from one to the maximum allowable value that corresponds to a determined neural network to find the best combination. The neural network with the smallest testing error was selected as the translator. Figures 4.17 to 4.20 show the relation between standard deviations of training and testing results and the number of design variables in the neural network for the I/O, F/A, torsion stiffnesses and mass of the B-pillar to rocker joint. 4.5 Results and Discussion Figures 4.17-4.20 compare the results obtained using different translators. The best results from the alternate translators are listed in Table 4.8. Figure 4.24 compare the predictions of translators A and FEA results. The following discuss the results corresponding to the stiffnesses in each direction and the mass. 4.5.1 Comparison of Predictions from Translators A and FEA Results We compared the relation between the stiffness and four most important design variables (outborad_cell_with, door_edge_width, thickness of frontrock, and A7) as predicted from translators A and FEA. Figure 4.24 shows that the predictions from both RSP and NN translators have the same trends as the FEA results. Both RSP and NN results correlate well with FEA results. 80 4.5.2 I/O Stiffness Table 4.8 and Figure 4.17 compare the results of polynomials and neural networks for different numbers of design variables. It is observed that the standard deviation of fitting and testing results of the polynomial decreases with the number of design variables when this number is less than 13. For more than 13 variables, the standard deviations of the fitting and testing results are practically constant as the number of design variables increases. Since it is difficult to choose the appropriate second degree polynomial model on the basis of the standard deviation only, we use two other criteria, namely, Cp statistic and AIC to help determine which model has the best generalization performance (see Chapter 2.4.3 for the definitions of Cp statistic and AIC criterion). Based on the results of the Cp statistic and the AIC criterion (Section 4.5.6), we decide to use the second degree polynomial model with 24 design variables as the polynomial translator for the I/O stiffness. It is observed that the neural network is more accurate than the polynomial according to the fitting and testing results. However, the difference between the neural network and the polynomial is very small in testing. Moreover, the accuracy of the neural network deteriorates more than the accuracy of the polynomial when we compare the training and testing results. This indicates that the polynomial is more robust than the neural network, probably because the latter has more unknown parameters than the former. The neural network with 24 design variables has the smallest standard deviation for the testing results, and is chosen as the neural network translator for the I/O stiffness. 4.5.3 F/A Stiffness Table 4.8 and Figure 4.18 show the standard deviations of both fitting and testing results for the F/A stiffness. The error in the F/A stiffness is almost 50% smaller than that in the I/O stiffness. It is observed that testing results are at least as accurate as fitting results for the polynomial. This indicates that sufficient designs have been used to fit the polynomial, and the second degree polynomial model does not have any unimportant (redundant) design variables. When we consider more than 5 design variables in the 81 regression model, the standard deviations of the testing results fall below those of the fitting results. We think that this is due to the randomness of the designs that we used for fitting and testing. Even if we only consider three variables in the regression model, the standard deviation of the testing results fall below 10%. Comparing the results from the linear polynomial model and from second degree polynomials (Table 4.8), it is found that the linear regression model is slightly more accurate. The reason is that the linear polynomial has more design variables than the second degree polynomial. The AIC criterion leads to the same conclusion (Figure 4.29). Therefore, the linear polynomial model with 21 design variables is chosen as the polynomial translator. The neural networks are almost equally accurate as the second degree polynomials. It is found that the errors of fitting and testing results consistently decreases as the number of design variables increases. We did not consider the second degree polynomial and the neural network with more than 12 designs because the polynomial and neural network predictions are accurate enough compared with the prediction for the I/O stiffness. The neural network with 12 design variables gives the best prediction for the testing designs, and is chosen as the NN translator for F/A stiffness. 4.5.4 Torsion Stiffness Table 4.8 and Figure 4.19 show the standard deviations for the fitting/training and testing results for the torsion stiffness. The error is smaller than that in I/O stiffness but larger than that in F/A stiffness. It is also observed that the polynomial and the neural network are almost equally accurate in terms of the testing results. Figures 4.19 and 4.18 look similar. The only difference is that when we include more design variables in the second degree polynomial, the standard deviations of the testing results are larger than those of the fitting results for the torsion stiffness, while for F/A stiffness, the testing curve is below the fitting curve. Since both the second degree polynomial and the neural network with 12 design variables gave the best predictions, they were chosen as the polynomial and the neural network translators for torsion stiffness. 82 4.5.5 Mass The relation between the design variables and the mass is the simplest. From Figure 4.20 and Table 4.8, it is observed that increasing the number of design variables in the polynomial consistently decreases the standard deviations of both fitting and testing results. Since in the future, mass will be used as the objective function when we optimize the joint, we want to include as many design variables as possible. Table 4.8 shows that the linear polynomial is more accurate than the neural network and the second degree polynomial. The Cp statistic and the AIC criterion also showed that the linear model should be the best model (see section 4.5.6 for details). Therefore, we selected the linear polynomial with 31 design variables as the translator A for the mass. The neural network with 12 design variables gives the best prediction compared with the other neural networks. Therefore, it is chosen as the NN translator for mass. 4.5.6 Validation The polynomial regression models obtained using stepwise regression were checked using the CP criterion and the AIC criterion. Figures 4.25-4.28 show the results of validating the regression model selected in Section 4.5.2-4.5.5 using the CP criterion. Figures 4.29-4.32 show the results when we use the AIC criterion to validate previous regression models. CP Criterion For the I/O stiffness, the CP criterion was used to check the linear polynomial model and the second degree polynomial model using 24 design variables. The results are shown in Figure 4.25. The linear and the second degree polynomial models selected in Section 4.5.2 have 23 and 24 design variables, respectively. It is observed that the value of CP for the selected polynomials is very close to the value of p, which is the number of parameters in the polynomial. Therefore, according to CP criterion, both regression models are suitable for predicting the I/O stiffness. 83 Figure 4.26 shows the results for the F/A stiffness of the B-pillar to rocker joint for the linear polynomial model and the second degree polynomial model with 12 design variables. The two models have 22 and 31 terms, respectively. It is observed that both models are appropriate according to the CP criterion. Figure 4.27 shows the results for the torsion stiffness. It is observed that both the linear regression model and the second degree polynomial model with 12 design variables are appropriate according to the CP criterion. Figure 4.28 shows the values of CP versus p for the mass of the B-pillar to rocker joint. The CP values for both the linear regression model and the nonlinear regression model with 12 design variables are very close to the number of parameters in the polynomials. However, the value of CP is larger than the number of terms for the second degree polynomial. This indicates that the second degree polynomial does not have enough terms. However, we did not try to improve this polynomial because its error is already much smaller than the errors corresponding to the stiffnesses in the other directions. AIC Criterion Figure 4.29 shows the relation between the AIC values and number of terms in the polynomial, p, for both the linear polynomial model and the second degree polynomial model for the I/O stiffness. It is observed that the nonlinear regression model with 24 design variables (the polynomial has 54 terms) has the lowest AIC value. According to AIC criterion, the second degree polynomial with 24 design variables is the best polynomial model. This observation is consistent with the results of Table 4.8. The relations between the AIC values and p of two models for F/A stiffness are compared in Fig. 4.30. The linear polynomial and the second degree polynomial have 21 and 12 design variables, respectively. It is observed that the linear regression model has 84 lower AIC value compared with the nonlinear regression model. This indicates that the linear regression model has better generalization performance than the second degree polynomial. This is also consistent with the results in Table 4.8. Figure 4.31 shows that the linear polynomial model and the second degree polynomial with 12 design variables have almost equal AIC value. This shows that the two models have almost same accurate performance. Again, this observation agrees with the results in Table 4.8. Figure 4.32 shows that, for the mass of the B-pillar to rocker joint, the linear polynomial has much small AIC value compared with the second degree polynomial that has 12 design variables. According to AIC criterion, the linear regression model is more accurate than the second degree polynomial. Table 4.8 leads to the same conclusion: the standard deviation of the linear polynomial is 0.6%, while the standard deviation of the second degree polynomial is 1.4%. 4.5.7 Conclusions In this study, the procedure of constructing translators A for the B-pillar to rocker joint was explained. Several polynomial models were compared. The CP criterion and the AIC criterion were used to validate obtained results. Neural networks were also used as an alternative to simulate the mapping relation between the design variables and the stiffness and mass of the joint. The results from different models were compared. Both polynomials and neural networks simulated FEA results quite accurately. It was observed that the neural networks and the polynomials were almost equally accurate. However, the neural networks deteriorated more than the polynomials when they predicted the stiffness of designs that they had not seen in training. Using the CP and AIC criteria to compare models, we obtained results that were consistent with those from using the standard deviation of the ratio of predictions over FEA results. 85 Table 4.1: Comparison of Stiffness and Mass for Different Types of Rocker Cross Section Cross Section I/O F/A Torsion Mass Type (Nmm) (Nmm) (Nmm) (Nmm) Car A Generic 4.5456E7 5.3119E8 8.7844E7 5.9589 Non-Generic 4.5709E7 5.3122E8 8.7850E7 6.1517 Car B Generic 9.1601E6 2.1839E8 5.3031E7 5.0346 Non-Generic 9.1488E6 2.1838E8 5.3040E7 5.1210 baseline Generic 3.3169E7 3.4940E8 8.1886E7 5.5694 Non-Generic 3.3168E7 3.4932E8 8.1887E7 5.6927 Cars A and B are the same as the cars 1 and 2 in Table 4.2. Baseline is the joint whose dimensions are equal to the mean values of dimensions measured in Table 4.2. Cars Table 4.2: Measured Dimensions for B-pillar to Rocker Joint Name of Design Variables Length_of_flange Spot_weld_spacing Spot_weld_placement Pillar_base Pillar_angle Pillar_io_angle Pillar_height Pillar_location Outer_pillar_width Inner_pillar_width Pillar_outer_length Pillar_inner_length Fwd_inner_ver_blending_rad Fwd_inner_hor_blending_rad Fwd_outer_ver_blending_rad Fwd_inner_hor_blending_rad Aft_inner_ver_blending_rad Aft_inner_hor_blending_rad Aft_outer_ver_blending_rad Aft_outer_hor_blending_rad Inner_pillar_base_width Rocker_length Pillar_reinforcement_depth Pillar_reinf_extended Pillar_reinf_base_width Car 1 17 45 8 185 90 90 220 143 70 10 90 122 130 125 130 125 105 105 117 115 15 490 100 Yes 60 Dimensions (mm, degree) Car 2 Car 3 Car 4 Min 18 20 20 17 40 50 55 40 7 8 8 7 157 215 180 157 90 90 90 90 90 90 90 90 210 250 210 210 186 160 163 143 83 67 50 50 6 6 25 6 77 107 110 77 130 160 155 122 124 155 120 120 118 80 90 80 124 155 120 120 118 80 90 80 120 133 95 95 135 115 100 100 120 140 95 95 135 130 100 100 12 2 2 480 483 480 480 80 80 80 Yes No Yes 30 20 20 Max 20 55 8 215 90 90 250 186 83 25 110 160 155 125 155 125 133 135 140 135 15 490 100 60 86 Pillar_reinf_expansion A1 A2 A3 A5 A7 A8 Frontrock_generic_type Rocker_height Inner_flange_distance Inner_rocker_height Rocker_width Outboar_cell_width Door_edge_height Door_edge_width Low_door_ht_minus_clearane Fwd_bulk_head_position Aft_bulk_head_position Outboard_rocker_bulkheads Inboard_rocker_bulkheads Top_pillar_hole Bottom_pillar_hole Fwd_pillar_hole Aft_pillar_hole Thickness of frontrock Thickness of pillar_reinf Thickness of pillarback Thickness of backrock Thickness of centerplate Thickness of pillar_bridge Thickness of bulkheads 10 90 70 10 75 70 80 No 117 22 90 147 65 6 19 66 37 No Yes 22 35 17 15 0.71 0.71 0.89 1.78 0.71 1.78 15 90 65 11 80 86 85 No 117 18 96 130 74 21 16 75 10 470 No Yes 35 28 18 15 0.94 0.74 1.27 1.65 0.74 1.40 85 75 36 87 80 90 No 120 28 110 120 95 38 7 65 2. 83. 78 22 87 80 90 Yes 110 45 90 115 70 32 10 60 2. 83. 65 10 75 70 80 110 18 90 115 65 6 7 60 15 90 78 36 87 86 90 120 45 110 147 95 38 19 75 No No 80 50 30 20 1.07 1.52 1.07 1.98 1.07 No No 39 45 17 18 1.27 1.07 1.27 1.27 1.27 22 28 17 15 0.71 0.71 0.89 1.27 0.71 80 50 30 20 1.27 1.52 1.27 1.98 1.27 87 Table 4.3: Seq No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Ranges of Design Variables Used in Creating the Database Bounds (mm, degree) Lower Upper 19 19 47.5 47.5 8.5 8.5 186 215 90 90 90 90 230 250 164.5 186 66.5 83 15.5 25 93.5 110 141 160 137.5 155 102.5 125 137.5 155 102.5 125 114 133 117.5 135 117.5 140 117.5 135 8.5 15 485 485 90 100 40 60 8.5 15 86.5 90 71.5 78 23 36 81 87 78 86 85 90 115 120 31.5 45 100 110 131 147 80 95 22 38 13 19 67.5 75 470 470 10 10 51 80 Comments 1 2 3 4 5 Name of Design Variables Length_of_flange # Spot_weld_spacing # Spot_weld_placement # Pillar_base Pillar_angle # Pillar_io_angle # Pillar_height Pillar_location Outer_pillar_width Inner_pillar_width Pillar_outer_length Pillar_inner_length Fwd_inner_ver_blending_rad Fwd_inner_hor_blending_rad Fwd_outer_ver_blending_rad ## Fwd_inner_hor_blending_rad ## Aft_inner_ver_blending_rad Aft_inner_hor_blending_rad Aft_outer_ver_blending_rad ## Aft_outer_hor_blending_rad ## Inner_pillar_base_width Rocker_length # Pillar_reinforcement_depth ## Pillar_reinf_base_width Pillar_reinf_expansion A1_ A2_ A3_ A5_ A7_ A8_ Rocker_height Inner_flange_distance Inner_rocker_height Rocker_width Outboar_cell_width Door_edge_height Door_edge_width Low_door_ht_minus_clearane Fwd_bulk_head_position Aft_bulk_head_position Top_pillar_hole 6 7 8 9 10 11 88 43 Bottom_pillar_hole 39 50 44 Fwd_pillar_hole 23.5 30 45 Aft_pillar_hole 17.5 20 46 Thickness of frontrock 1 1.27 47 Thickness of pillar_reinf 1.11 1.52 48 Thickness of pillarback 1.08 1.27 49 Thickness of backrock 1.63 1.98 50 Thickness of centerplate 1 1.27 51 Thickness of pillar_bridge 1 1 52 Thickness of bulkheads 1.59 1.78 # fixed ## dependent 1 Length_of_flange is fixed at 19 mm. 2 Spot_weld_spacing is fixed at 47.5 mm. 3 Spot_weld_placement is fixed at the center of flange. 4 Pillar_angle is fixed at 90 degrees. 5 Pillar_io_angle is fixed at 90 degrees. 6 Fwd_outer_ver_blending_rad is assumed to be equal to Fwd_inner_ver_blending_rad. 7 Fwd_outer_hor_blending_rad is assumed to be equal to Fwd_inner_hor_blending_rad. 8 Aft_outer_ver_blending_rad is assumed to be equal to Aft_inner_ver_blending_rad. 9 Aft_outer_hor_blending_rad is assumed to be equal to Aft_inner_hor_blending_rad. 10 Rocker_length is fixed at 485 mm. 11 Pillar_reinforcement_depth can be derived from other design variables. 89 Table 4.4: Ranking of Important Dimensions for I/O Stiffness Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Dimensions Outboard_cell_width Door_edge_width Thickness of frontrock2 A7 Pillar_base Inner_pillar_base_width Door_edge_height Thickness of backrock2 Aft_inner_ver_blending_rad Pillar_location Rocker_width A5 Thickness of pillarback2 Inner_rocker_height Pillar_inner_length A2 Bottom_pillar_hole Fwd_inner_hor_blending_rad Outer_pillar_width Aft_inner_hor_blending_rad Thickness of pillar_reinf Low_door_ht_minus_clearance Pillar_height Fwd_inner_ver_blending_rad 90 Table 4.5: Ranking of Important Dimensions for F/A Stiffness Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Dimensions Thickness of frontrock2 Pillar_inner_length Pillar_base Thickness of pillarback2 Bottom_pillar_hole Outer_pillar_width Pillar_outer_length Thickness of backrock2 Top_pillar_hole Aft_inner_ver_blending_rad Pillar_height Fwd_pillar_hole Rocker_width Outboard_cell_width Door_edge_height Door_edge_width A5 Thickness of pillar_reinf Low_door_ht_minus_clearance A8 Table 4.6: Ranking of Important Dimensions for Torsion Stiffness Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Dimensions Outer_pillar_width Thickness of frontrock2 Pillar_height Outboard_cell_width Top_pillar_hole Pillar_base Inner_pillar_base_width Door_edge_width Fwd_pillar_hole Thickness of pillarback2 Pillar_inner_length Inner_pillar_width Aft_inner_ver_blending_rad Door_edge_height Aft_pillar_hole A7 Fwd_inner_hor_blending_rad Bottom_pillar_hole Pillar_location Pillar_reinf_base_width 91 Table 4.7: Ranking of Important Dimensions for Mass Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Dimensions Thickness of frontrock2 Thickness of pillar_reinf Thickness of backrock2 Thickness of centerplate Rocker_width Pillar_height Thickness of pillarback2 Inner_flange_distance Rocker_height Outer_pillar_width Pillar_base A2 A5 Pillar_inner_length Inner_rocker_height A3 Pillar_outer_length Pillar_reinf_base_width Top_pillar_hole Outboard_cell_width Table 4.8: Comparison of Different Models for B-pillar to Rocker Joint Stiffness/ Mass I/O NN Polynomial F/A NN Polynomial Torsion NN Polynomial Mass NN Type Polynomial Model Linear 2nd degree Linear 2nd degree Linear 2nd degree Linear 2nd degree No of Variables 23 24 24 21 12 12 23 12 12 31 12 12 Std deviation Fitting/Training 0.1354 0.0880 0.0755 0.0508 0.0519 0.0533 0.0716 0.0647 0.0609 0.0062 0.0124 0.0127 Std deviation Testing 0.1182 0.0982 0.0872 0.0447 0.0518 0.0485 0.0745 0.0720 0.0716 0.0063 0.0138 0.0139 Note: Only the best second order polynomial results and the best neural network results are listed. 92 B-pillar Front Outside Rocker Figure 4.1a: B-pillar to Rocker Joint 93 Torsion Stiffness I/O Stiffness F/A Stiffness Front Outside Fixed Fixed Figure 4.1b: Definition of Stiffness for B-pillar to Rocker Joint 94 Pillar back Pillar reinforcement Front Outside Back rocker Front rocker Center plate Figure 4.2: Parts of B-pillar to Rocker Joint 95 Front Outside Pillar Reinforcement Figure 4.3a: Extended B-pillar Reinforcement: 96 Front Outside Pillar Reinforcement Figure 4.3b: Non-Extended B-pillar Reinforcement: 97 B-pillar Front Outside Rocker Pillar bridge Figure 4.4: Pillar Bridge Reinforcement 98 B-pillar Front Outside Rocker Bulkheads Figure 4.5: Bulkhead Reinforcement 99 Type 1: Generic Type Type 2: Non Generic Type Figure 4.6: Two Different Types of Rocker Cross Section 100 z” z’ O’z” // Oz y’A x’ y’A z’ x’ Pillar_angle z y’ O’ O y Pillar_io_angle Figure 4.7*: B-pillar Orientation 101 Door_edge_width A5 Inner_ rocker_ height Rocker_ height A8 A1 A7 Door_edge_height Low_door_ht_ minus_clearance A3 A2 Inner_flange_distance Outboard_cell_width Rocker_width Figure 4.8: Rocker Cross Section 102 Pillar_inner_length Inner_pillar_width Outer_pillar_width Pillar_outer_length Figure 4.9*: B-pillar Dimensions 103 Fwd_inner_hor_blending_rad Aft_inner_hor_blending_rad Fwd_inner_blending_rad Aft_inner_ver_ blending_rad Pillar_ height Fwd_outer_ ver_blending_ rad Pillar_base Aft_outer_ver_ blending_rad Pillar_location Fwd_outer_hor_ blending-rad Aft_outer_hor_ blending_rad Rocker_length Figure 4.10*: B-pillar to Rocker Blending Radii and Rocker Dimensions 104 Pillar_base Pillar_ reinforcement_ depth Pillar_reinf_ expansion Pillar_reinf_ base_width Figure 4.11*: Extended Pillar Reinforcement 105 Top_pillar_hole Aft_pillar_hole Fwd_pillar_hole Bottom_pillar_hole Figure 4.12*: Opening in Back of Pillar 106 Flange Spot_weld_placement Flange Spot_weld_Spacing Length_of_flange Figure 4.13: Flanges and Spot Welds 107 B-pillar Front Outside Fwd_bulk_head_position Rocker Aft_bulk_head_position Figure 4.14: Dimensions for Bulkhead 108 Pillar back Inner_pillar_base_width Front rock Center plate Pillar reinforcement Back rocker Pillar_reinf_base_width Figure 4.15: Dimensions for Back Rocker and Pillar Reinforcement 109 Y Z X Figure 4.16: FEA Mesh of B-pillar to Rocker Joint 110 0.35 Comparison Between Polynomial and Neural Network Results (I/O) 0.30 Standard Deviation 0.25 0.20 Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing 0.15 0.10 0.05 0 5 10 15 20 25 Number of Design Variables Figure 4.17: Comparison of Polynomial and Neural Network Results for I/O Stiffness 111 Comparison Between Polynomial nad Neural Network Results (F/A) 0.140 0.130 0.120 0.110 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0 2 4 6 8 10 12 Standard Deviation Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing Number of Design Variables Figure. 4.18: Comparison of Polynomial and Neural Network Results for F/A Stiffness 112 Comparison Between Polynomial and Neural Network Results (Torsion) 0.25 0.20 Standard Deviation Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing 0.15 0.10 0.05 0 2 4 6 8 10 12 Number of Design Variables Figure 4.19: Comparison of Polynomial and Neural Network Results for Torsion Stiffness 113 0.070 Comparison Between Polynomial and Neural Network Results (Mass) 0.060 0.050 Standard Deviation Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing 0.040 0.030 0.020 0.010 0.000 0 2 4 6 8 10 12 Number of Design Variables Figure 4.20: Comparison of Polynomial and Neural Network Results for Mass 114 Comparison of Two Different Normalization Methods 0.40 0.35 Standard Deivation of Testing 0.30 Normalized to [-1,+1] Divided by mean 0.25 0.20 0.15 0.10 0 10 20 30 40 2 50 60 70 80 Time of Training (×10 epochs) Figure 4.21: Comparison of Two Different Normalization Methods 115 Relation Between Standard Deviations and Time of Training 0.16 Training Testing 0.14 Standard Deviation 0.12 0.10 0.08 0.06 0 10 20 2 30 40 Time of Training ( ×10 epochs) Figure 4.22: Relation Between Standard Deviation and Time of Training 116 0.16 Relation Between Standard Deviation and Number of Neurons (I/O) Relation Between Standard Deviation and Number of Neurons (I/O) 0.16 0.14 0.14 Standard Deviation of Testing Results 0.12 Use 6 parameters Standard Deviation of Testing Results 0.12 0.10 0.10 Use 12 parameters 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0 10 20 30 40 0.00 0 5 10 15 20 25 Number of Neurons in the Hidden Layer Number of Neurons in the Hidden Layer Relation Between Standard Deviation and Number of Neurons (I/O) 0.16 Relation Between Standard Deviation and Number of Neurons (I/O) 0.160 0.14 0.140 Standard Deviation of Testing Results Standard Deviation of Testing Results 0.12 0.120 0.10 0.100 Use 13 parameters 0.08 0.080 Use 24 parameters 0.06 0.060 0.04 0.040 0.02 0.020 0.00 0 5 10 15 20 0.000 0 2 4 6 8 10 12 Number of Neurons in the Hidden Layer Number of Neurons in the Hidden Layer Figure 4.23: Relation Between Standard Deviation and Number of Neurons 117 Comparison of FEA, Polynomial and Neural Network Results (I/O) 5.5000E7 Comparison of FEA, Polynomial and Neural Network Results (I/O) 5.0000E7 5.0000E7 4.5000E7 FEA Results Polynomial Results (Use 13 parameters) Polynomial Results (Use 24 parameters) NN Results (Use 13 parameters) NN Results (Use 24 parameters) 4.5000E7 FEA Results Polynomial Results (Use 13 parameters) Polynomial Results (Use 24 parameters) NN Results (Use 13 parameters) NN Results (Use 24 parameters) 4.0000E7 I/O Stiffness 4.0000E7 3.5000E7 3.0000E7 2.5000E7 2.5000E7 2.0000E7 65 I/O Stiffness 3.5000E7 3.0000E7 2.0000E7 70 75 80 85 90 95 7 8 9 10 11 12 13 14 15 16 17 18 19 Column 36: outboard cell width Column 38: door edge width Comparison of FEA, Polynomial and Neural Network Results (I/O) 4.5000E7 3.9000E7 Comparison of FEA, Polynomial and Neural Network Results (I/O) FEA Results Polynomial Results (Use 13 parameters) Polynomial Results (Use 24 parameters) NN Results (Use 13 parameters) NN Results (Use 24 parameters) 4.0000E7 FEA Results Polynomial Results (Use 13 parameters) Polynomial Results (Use 24 parameters) NN Results (Use 13 parameters) NN Results (Use 24 parameters) 3.8000E7 3.7000E7 3.6000E7 I/O Stiffness I/O Stiffness 0.8 0.9 1.0 1.1 1.2 1.3 3.5000E7 3.5000E7 3.4000E7 3.3000E7 3.2000E7 3.0000E7 2.5000E7 3.1000E7 3.0000E7 2.0000E7 0.7 2.9000E7 70 72 74 76 78 80 82 84 86 Column 46: thickness of frontrock Column 30: A7 Figure 4.24: Comparison of FEA Results and Predictions of RSP and NN Translators for the I/O Stiffness of B-pillar to Rocker Joint 118 Relation Between Cp and p (For the Linear Model of I/O Stiffness) 200 Cp 150 Cp 100 50 0 0 5 10 15 20 p (Number of Parameters in Polynomial) 200 Relation Between Cp and p (For the Nonlinear Model of I/O Stiffness Using 24 Design Variables) 150 Cp Cp 100 50 0 0 10 20 30 40 50 p (Number of Parameters in Polynomial) Figure 4.25: Relation Between Cp and p for I/O Stiffness 119 Relation Between Cp and p (For the linear Model of F/A) 200 150 Cp Cp 100 50 0 0 5 10 15 20 p (Number of Parameters in Polynomial) 200 Relation Between Cp and p (For the Nonlinear Model of F/A Using 12 Design Variables) Cp 150 Cp 100 50 0 0 5 10 15 20 25 30 p (Number of Parameters in Polynomial) Figure 4.26: Relation Between Cp and p for F/A Stiffness 120 200 Relation Between Cp and p (For the Linear Model of Torsion) Cp 150 Cp 100 50 0 0 5 10 15 20 p (Number of Parameters in Polynomial) 200 Relation Between Cp and p (For the Nonlinear Model of Torsion Stiffness Using 12 Design Variables) 150 Cp Cp 100 50 0 0 5 10 15 20 25 30 p (Number of Parameters in Polynomial) Figure 4.27: Relation Between Cp and p for Torsion Stiffness 121 Relation Between Cp and p (For the Linear Model of Mass) 200 Cp 150 Cp 100 50 0 0 5 10 15 20 25 30 p (Number of Parameters in Polynomial) 200 Relation Between Cp and p (For the Nonlinear Model of Mass Using 12 Design Variables) 150 Cp Cp 100 50 0 0 5 10 15 20 p (Number of Parameters in Polynomial) Figure 4.28: Relation Between Cp and p for Mass 122 Relation Between AIC and p for I/O Stiffness 1.2400E4 linear model of I/O Stiffness nonlinear model of I/O Stiffness 1.2300E4 using 13 design variables nonlinear model of I/O Stiffness using 24 design variables 1.2200E4 AIC final point final point 1.2100E4 final point 1.2000E4 1.1900E4 0 10 20 30 40 50 60 70 p (Number of Parameters in Polynomial) Figure 4.29: Relation Between AIC and p for Different Regression Models for I/O Stiffness 123 Relation Between AIC and p for F/A Stiffness 1.3800E4 1.3700E4 linear model of F/A stiffness nonlinear model of F/A stiffness using 12 design variables 1.3600E4 AIC 1.3500E4 final point 1.3400E4 final point 1.3300E4 0 10 20 30 40 50 60 p (Number of Parameters in Polynomial) Figure 4.30: Relation Between AIC and p for Different Regression Models for F/A Stiffness 124 Relation Between AIC and p for Torsion Stiffness 1.3800E4 1.3600E4 linear model of torsion stiffness 1.3400E4 nonlinear model of torsion stiffness using 12 design variables 1.3200E4 AIC 1.3000E4 1.2800E4 1.2600E4 final point final point 1.2400E4 1.2200E4 0 10 20 30 40 50 60 70 p (Number of Parameters in Polynomial) Figure 4.31: Relation Between AIC and p for Different Regression Models for Torsion Stiffness 125 Relation Between AIC and p for Mass -1700 -1800 -1900 -2000 -2100 linear model of mass nonlinear model of mass using 12 design variables AIC -2200 final point -2300 -2400 final point -2500 -2600 -2700 0 10 20 30 40 50 60 70 p (Number of Parameters in Polynomial) Figure 4.32: Relation Between AIC and p for Different Regression Models for Mass 126 Chapter 5 Developing Translator A for the A-pillar to Roof Rail Joint 5.1. Introduction This chapter applies the general methodology for developing translator A to the Apillar to roof rail joint. The A-pillar to roof rail joint is a “Y” shape joint (Figs. 1.1, 5.1a and 5.1b), located on the upper left and upper right sides of the driver and the passenger, respectively. The A-pillar joint supports the windshield of a car, the roof and part of the front door. The stiffness of the A-pillar to roof rail joint has a big effect on the safety in crash and the overall stiffness of the car body. Section 5.2 describes the geometry of the A-pillar to roof rail joint. This section also describes a parametric Pro/Engineer model of this joint, and the procedure for FEA of the A-pillar to roof rail joint. Section 5.3 explains how we created a database for developing translator A. Section 5.4 describes the procedure for developing translator A using response surface polynomials and neural networks. Section 5.5 discusses the polynomial and neural network results. 5.2 Description of the A-pillar to Roof Rail Joint The A-pillar to roof rail joint has three branches. The branch that is perpendicular to the side of a car is called header. The branch that is parallel to the top of the front door is called roof rail. A-pillar is the branch that lies between the windshield and the front door. 127 The header branch and the roof rail branch are generally perpendicular to each other. The A-pillar branch is inclined at an angle relative to the roof surface (Fig. 5.1a and 5.1b). Most A-pillar to roof rail joints include four parts (Fig. 5.2): • • • • Part 1 is the roof plate. Part 2 is the outer cell of the joint. Part 3 is the inner shell of the joint. Part 4 is the lower shell of the header. To increase the stiffness of the joint, most A-pillar joints also have reinforcements. The types of reinforcements vary greatly from one joint to another. By examining the hardware joints, we classified reinforcements into the following three types: • Type 1: A reinforcement that connects the two flanges at A-pillar and roof rail cross sections (Fig. 5.3). This reinforcement is called part 5 in this study. • Type 2: A reinforcement that follows the shape of the top plate of the header and is connected to the bottom piece of header (part 4) to create a closed box (Fig. 5.3). This reinforcement is called part 6. • Type 3: A reinforcement that follows the shape of the top plate of the A-pillar and roof rail (Fig. 5.2). This type of reinforcement is called part 7. The A-pillar to roof rail hardware joints examined in this study had considerably different types of construction. The parametric model built in this study was based on some simplifications. The architecture of the parametric model was selected in consultation with engineers of an automotive company. This is close to the architecture of an actual car joint. The parametric model in this study can account for all three types of reinforcements mentioned above. The parametric model was created using Pro/Engineer. The parametric model is generic, and can be easily modified by changing the values of its design variables. 128 As explained in Chapter 2, we divided the design variables of a joint into six levels. The higher levels define the overall joint shape while the lower levels define the details of the joint geometry. The following sections describe these variables. First, we explain the higher level variables that define the orientation and position of each branch. Then, we define the variables that determine the cross section of each branch. Finally, the design variables defining the blending areas are explained. 5.2.1 Design Variables Defining the Orientation and Position of the Branches To complete describe the geometry of the A-pillar to roof rail joint, we first define a global coordinate system (Fig. 5.4). The origin of the global coordinate system, O, is the intersection of lines tangent to the outboard flange of the A-pillar section and the outboard flange of the roof rail section. The x-axis of the global coordinate system is toward the outboard side of the car. The y-axis is vertical. The z-axis is toward the front direction of the car. All local coordinate systems at the cross sections of the three branches are defined relative to the global coordinate system. To define the cross section of a branch, we first define a local coordinate system for that branch. The x-axis of this coordinate system is parallel to the axis of this branch, and its y-axis is parallel to the direction of one flange on the cross section of this branch. The z-axis is the cross-product of x and y axes. Each vertex on this cross section is defined relative to the origin of this local coordinate system. As explained in Section 5.2, the header and the roof rail are perpendicular to each other. The axes of these two branches are defined to be parallel to x-axis and z-axis of the global coordinate system (Fig. 5.4). 129 The origin of the local coordinate system of the header branch is defined using its three coordinates relative to the global coordinate system. They are called header_offest, header_vertical_offset, and header_horizontal_offset (Figs. 5.5 and 5.6). The x-axis of the roof rail branch coincides with the z-axis of the global coordinate system, so only one dimension is needed to define the local coordinate system of roof rail branch. This is called roof_rail_offset (Figs. 5.5 and 5.6). To define the orientation and position of the A-pillar branch, we use two angles and one length (Figs. 5.5 and 5.6), namely, theta, phi, and a_pillar_offset. The two angles define the orientation of A-pillar relative to the global coordinate system. A_pillar_offset is the distance between the origin of the global coordinate system and that of the local coordinate system. 5.2.2 Dimensions Defining the Cross Section of the Header The y-axis of the local coordinate system of the header is assumed horizontal, that is, parallel to the z-axis of the global coordinate system. Flange 1 and Flange 3 (Fig. 5.8) are also assumed parallel to the z-axis of the global coordinate system. Also, the upper and lower plates of the header are assumed horizontal. In general, we need 2(nedge-1) design variables to define a cross section made up of straight plates, where nedge is the number of edges. The header has seven edges. Therefore, we need 12 design variables. The assumption mentioned in the previous paragraph reduces the number of variables from 12 to eight. The dimensions that define the cross section of the header have the prefix “H_”. Eight dimensions (Fig. 5.8) are needed to completely define the cross section of the header. 130 5.2.3 Dimensions Defining the Cross Section of the Roof Rail The top plate of the roof rail branch is assumed to be parallel to the x-axis of the global coordinate system. Flange 3 is also assumed to be parallel to the x-axis of the global coordinate system. The dimensions used to define the cross section of roof rail have the prefix “RR_”. Twelve dimensions are needed to completely describe the geometry of the roof rail section (Fig. 5.9. Flange2_width and flange3_width are not shown). 5.2.4 Dimensions Defining the Cross Section of the A-pillar The cross section of the A-pillar is the most difficult one to parameterize. Two sets of dimensions are used to describe the geometry of the A-pillar cross section. The set of dimensions that have prefix “AP_” define the outer plate shell of the A-pillar section (Fig. 5.10). The other set of dimensions with the prefix “AP_reinf_” define the reinforcement of the A-pillar (Fig. 5.11). Eighteen dimensions (dimensions flang1_width and flange2_width are not shown in Figs. 5.10 and 5.11) are used for the cross section of the A-pillar. 5.2.5 Blending Radii Three dimensions, namely, H_blending_rad, RR_blending_rad, and AP_blending_rad, define the blending radii of the joint (Fig. 5.5 and 5.6). Since a joint is created by first extruding the cross section of each branch, and then connecting the cross sections of two branches using blending curves, the extrusion distance of a branch is equal to the value of the offset distance of this branch subtracted by its blending radius. Therefore, the blending radii also determine the extrusion distances of the three branches. 5.2.6 Connections Ten dimensions are used to define the flanges and the spot welds. These dimensions include the dimensions that define the widths of flanges like flange1_width, the dimension defining the location of spot welds on the flange (distance_from_edge), the 131 dimension that defines the distance between two adjacent spot welds (spot_weld_spacing), and the angles between the flanges and their adjacent plates like flange1_angle_down and flange1_angle_up (Fig. 5.7-5.10). 5.2.7 Other Dimensions Dimensions part2_cut_distance, part3_cut_distance, part4_extension, and part7_cut_distance complete the definition of the geometry of parts 2-4 and 7 (Figs.5.125.15). 5.2.8 Parametric Model and FEA of the A-pillar to Roof Rail Joint After consulting engineers from an automotive company, we selected one type of construction to develop translator A. This type has the four parts that most A-pillar joints have, e.g., parts 1 to 4. Only one type of reinforcement, part 7, was considered in developing translator A. Parts 5 and 6, which were found in only a few joints, were not considered in the parametric model for developing translator A. A parametric model of the A-pillar to roof rail joint was developed based on the above variables. Pro/Engineer was used. The model has 48 independent variables, and can simulate any feasible joint design by changing the values of the design variables of these models. The Pro/Engineer model can account for packaging, manufacturing, and styling constraints, and can warn the user if a constraint is violated. After developing the geometric model corresponding to a joint design, we use Pro/Engineer to generate a FEA model and a NASTRAN bulk data file. The size of mesh in Pro/Engineer controls the size of the elements in the FEA model. We studied the effect of the mesh size on the FEA results of a design. Table 5.1 shows the results. It is observed that the mesh size affects slightly the FEA results. The difference for models with same geometry but different mesh size is less than 3%. A FEA model with smaller 132 mesh size tends to give smaller predictions for the I/O, F/A, torsion stiffness, and mass than models with larger mesh size. We also compared the predictions of the stiffnesses of two joints of two cars, called car A and car B, from the Pro/Engineer model with the experimental results (Table 5.2). It is observed that the FEA model predicts the I/O and torsion stiffnesses more accurately than the F/A stiffness. The errors of predictions of I/O and torsion stiffnesses for car A are –4.0% and –5.9%, respectively. For car B, the errors are 5.9% and –19.1%, respectively. The error in FEA result for F/A stiffness is 45.3% for one car and–15.8% for the other car. The prediction of F/A stiffness for the first car is big compared with predictions of stiffness in the other directions. This is due to several reasons. a) The Pro/Engineer model used in this study simplifies the actual joint hardware. Some details such as access holes are neglected. Moreover, part 7 in the parametric model is assumed to follow the shape of the outer shell of the A-pillar, which simplifies the actual part 7. b) In experiments, when measuring the F/A stiffness, engineers apply a moment in the horizontal direction on the cross section of A-pillar. Then, they measure the rotation of A-pillar in the horizontal direction, and calculate the stiffness using the following expression. Moment in the given direction Resulting rotation in same direction Stiffness = (5.1) Since in practice, it is difficult to define the horizontal direction of the A-pillar section, the actual moment and rotation may not be in the horizontal direction. As a result, the measured F/A stiffness is different from that in the model. Because F/A stiffness is sensitive to the direction, this also contributes to the difference between the predicted and the measured F/A stiffness for one car. 133 5.3 Developing a Database To develop translator A, we first develop a database. The database stores the values of dimensions of many designs and their performance characteristics (stiffness and mass) obtained using FEA. The input values of the design variables of a design should make a feasible Pro/Engineer model in order to use the parametric Pro/Engineer model to generate a MSC/NASTRAN bulk data file. Several methods can be used to create those designs, including Taguchi method, linear method, and random method. This study uses the random method to generate these designs as explained in Chapter 2. Before using a random number generator, we first measured the dimensions of six actual joints that were provided to us by an automotive manufacturer (Table 5.3). From these measurements we found the maximum and minimum values of each design variable. To account for more designs, we increased the ranges of design variables by 10% when developing the database (Table 5.4). Among all the design variables of the A-pillar joint, some are generally fixed because of manufacturing rules, such as the width of flanges like flange1_width, the location of spot welds on flanges (distance_from_edge) and the distance between two adjacent spot welds (spot_weld_spacing) (Fig. 5.7). Some dimensions are fixed because of styling requirements, such as the orientation of the A-pillar (theta and phi). To compare the results of different designs fairly, the lengths of the branches should be fixed. Unfortunately, there is not widely accepted rule for defining reference points with respect to which the lengths of the branches are measured. We found that the branches of the actual joints had different lengths. To circumvent this problem, the branches of the three branches were considered as variables and they varied when creating the database for translator A. 134 First, we used a random number generator to create random designs. The Pro/Engineer model checked if each design was feasible. Infeasible designs were screened out. Then using Pro/Engineer, the model corresponding to each feasible design was generated and a NASTRAN bulk data file was created. Figure 5.16 shows a FEA model of the A-pillar to roof rail joint. 600 models corresponding to feasible designs were developed. These models were analyzed using MSC/NASTRAN to obtain the stiffness and mass. The dimensions of all feasible designs and their performance characteristics were stored in a database that was later used to develop translator A. There were 600 designs in the database for the A-pillar to roof rail joint. 5.4 Developing Translator A Two different methods, namely, response surface polynomials and neural networks, were used to create translator A. As described in Chapter 2, the development of translator A involves several steps. First, the design variables are ranked in terms of their effects on the stiffness and mass. Then, different response surface polynomial models are tested. We choose the best regression model according to the results. Finally, neural networks with different architectures are considered, from which we choose the neural network translators. We also compare the response surface polynomial and neural network results. 5.4.1 Ranking Important Design Variables Many design variables are needed to completely describe the geometry of a joint. However, only a few affect significantly the stiffness and mass of the joint. It is also important for a designer to know the relative importance of each design variable so that the designer can determine the most effective way to improve the performance of a design. 135 The importance of design variables on the stiffness and mass was ranked using linear polynomial models. Stepwise regression was used in this study to find the best linear polynomial model. See Chapter 2 for details. Stepwise linear regression calculates the F-ratio of each parameter. F-ratio is the ratio of the decrease in the sum of squared errors of the results when a parameter is included in the regression model over the mean squared error of the final regression model. The F-ratio can be used as a measure of the importance of design variables. The design variable with bigger F-ratio has bigger effect on the output. According to the value of the F-ratio, the design variables are listed from the one with the biggest F-ratio to the one with the smallest F-ratio. See Tables 5.5 to 5.8 for the rankings of important design variables for I/O, F/A, torsion stiffness and mass of the A-pillar to roof rail joint. 5.4.2 Polynomial Translators We developed and tested three polynomial models, including linear polynomial models, second degree polynomial models, and double regression models. The following sections present the procedure for developing these models. Normalization of the Torsion Stiffness Before discussing different regression models, we first explain the normalization of the torsion stiffness. Figure 5.17 shows that the torsion stiffness correlated with the ratio of the thickness of part 3 over the offset of the A-pillar (Thickness of part 3 / A_pillar_offset). The correlation coefficient is 67.1%. By dividing the torsion stiffness by above quotient, we can reduce the range of the resulting normalized torsion stiffness. It is found that normalizing the torsion stiffness improves the fitting and testing results. Linear Polynomials A linear polynomial model was first obtained using stepwise regression. Only the important parameters were included in the regression model. Figure 5.18 shows the 136 scatter plots of the predictions of the linear model versus FEA results for the I/O, F/A, torsion stiffnesses and mass. There are 25 parameters (24 design variables) in the linear models for both I/O and F/A stiffnesses. The linear models for the torsion stiffness and mass have 29 and 32 parameters, respectively. The predictions from the linear polynomials were not satisfactory. The standard deviations of the ratio of predicted value over FEA result were 15.8% and 12.3% for the I/O and F/A stiffnesses, respectively. For the torsion stiffness and mass, the standard deviation of the fitting results were 13.9% and 2.2%, respectively. Second Degree Polynomials To improve the accuracy of the predictions of the translator, we also considered second degree polynomials. Figure 5.19 shows the scatter plots of the results obtained using second degree polynomials versus the FEA results for the I/O, F/A, torsion stiffnesses and mass. The results from the second degree polynomials were better than those from the linear polynomials (Tables 5.9-5.12). The standard deviations of fitting results for the I/O, F/A, torsion stiffnesses and mass were reduced to 13.7%, 10.6%, 13.8% and 2.0%, respectively. Double Regression models From Figure 5.18, it is observed that the predictions from the linear polynomial models are not evenly distributed along the two sides of a straight line inclined at 45 degrees relative to the horizontal axis. For very low values and very high values of the stiffness from FEA, the translator underestimates the stiffness, whereas for values of the stiffness from FEA close to the mean it overestimates the stiffness. Figure 5.20a shows the relation of the difference between the predictions and FEA results versus the FEA results for I/O stiffness. Figure 5.20b shows the predicted I/O stiffness from the linear 137 polynomial model versus the FEA results superimposed on a cubic polynomial. It is observed that the predicted results are almost evenly distributed along the two sides of the cubic polynomial. Based on such observation, a second regression is performed which uses a polynomial (cubic polynomials for I/O, F/A, quadratic polynomials for torsion and mass) to simulate the relation between the predicted results from the linear regression model and the FEA results. The following is the formulation for the second regression. ˆ ˆ y D = ∑ ci ( y L ) i i =0 nD (5.2) ˆ ˆ where y D and y L are the values of the dependent variables predicted from the double regression and the linear polynomial models, respectively. n D is the degree of the double regression model. n D =3 for the I/O and F/A stiffnesses, and n D =2 for the torsion stiffness and mass. Coefficients ci in Eq.5.2 are determined using regression. Figures 5.21a and 5.21b show the relations between the predictions from the double regression models and the FEA results for the fitting and testing results. It is found that the double regression model significantly improves the prediction compared with the linear regression model (Fig. 5.18). Moreover, the error is still significant for joints with high stiffness. The predictions of the translator are unbiased, with the exception of torsion stiffness. Explanation of Double Regression From Tables 5.9 to 5.12, we find that the double regression models are more accurate than the linear and second degree models. The following is an explanation. For simplicity, suppose that there are only two design variables in the linear polynomial model, and a quadratic polynomial is used in the second regression. The linear polynomial is: 138 ˆ y L = b0 + b1 x1 + b2 x2 (5.3) Substituting the above equation into the equation of double regression (Eq.5.2), we have ˆ ˆ ˆ2 y D = c0 + c1 y L + c2 y L = c0 + c1 b0 + c1 b1 x1 + c1 b2 x2 + c2 b02 + 2 2 c2 b12 x12 + c2 b2 x2 + 2c2 b0 b1 x1 + 2c2 b1 b2 x1 x2 + 2c2 b0 b2 x2 (5.4) = (c0 + c1 b0 + c b ) + (c1 b1 + 2c2 b0 b1 ) x1 + (c1 b2 + 2c2 b0 b2 ) x2 + 2 2 0 2 2 2c2 b1 b2 x1 x2 + c2 b12 x12 + c2 b2 x2 A second degree polynomial can be expressed as 2 ˆ yQ = q0 + q1 x1 + q2 x2 + q3 x1 x2 + q 4 x12 + q5 x2 (5.5) ˆ ˆ Comparing the expressions of y D and yQ , we find that that both expressions have same number of unknown parameters. It can be easily verified that if more design variables were considered in the linear regression model, there would be more unknowns ˆ in the expression of yQ . So theoretically, the quadratic regression should give better or at least equally good predictions for fitting. The reason is that if we consider a large number of design variables, say 24, we need to consider totally 325 (=( C 0 + 2C1 + C 2 ), C nj is the combination) parameters. Because this number is too V 24 24 24 high, we use a procedure that gradually increases the number of design variables when creating the second degree polynomials (see Section 2.4.2 for a detailed explanation). In each step, we use an incomplete second degree polynomial obtained from the previous step, and add a new design variable according to the ranking of important design variables. The new second degree polynomial includes the terms left from last regression, the new design variable, the square of the new variable, and interactions between the new variable and the variables of the previous model. We then use stepwise 139 regression to get a new second degree polynomial that considers more design variables. This method allows us to circumvent the limitation of the software. However, it sacrifices the accuracy of the second degree polynomial because it misses some terms that may be important. If all the combinations were considered in the regression model, a better results would be found (at least for fitting). Because of the software limitation and the method we use to circumvent this limitation, the double regression model gives better predictions than the quadratic regression model under most situations. 5.4.3 Neural Network Translators Neural Networks were also used to simulate the mapping relations between the design variables and the stiffness and mass of the A-pillar to roof rail joint. To develop the neural network translator we used the ranking of important design variables obtained using the linear regression models. First, only a few most important design variables were considered in the neural network according to the importance of each design variable. Then we gradually added more design variables to the neural network. For each neural network, the input design variables were normalized so that they could only vary in the range [-1, +1]. Chapter 4 showed that this approach worked better than either not normalizing the design variables or normalizing the design variables by their mean values. If we keep training the neural network over a very long period, we can improve the training results dramatically. However, the trained neural network has poor generalization performance, that is, it can not predict accurately the stiffness or mass of new designs that it has not seen in training. This problem is more frequent with neural networks with a large number of unknown parameters relative to the number of examples. We observed this problem when creating the translator A for the B-pillar to rocker joint. 140 To avoid this problem, cross-validation is used. Specifically, the 600 designs in the database are split into three groups: 300 designs are used for training; 100 designs are used to determine when to stop training; and the remaining 200 designs are used to test the generalization performance of the trained neural network. Cross-validation prevents a neural network from being over trained, improves the predictions of the trained neural network, and reduces the effect of the number of neurons in the hidden layers. Figures 5.21c and 5.21d show the scatter plots for the training and testing results of the neural network translators. 5.5 Results and Discussion Figures 5.22a to 5.22d and Tables 5.9 to 5.12 compare the results obtained using different methods. The following are observed regarding the stiffness in each direction and the mass. 5.5.1 I/O Stiffness Table 5.9 and Figure 5.22a compare the results from the linear polynomial, the second degree polynomials, the double regression model, and the neural networks. It is observed that neural networks give better training/fitting and testing results compared with the second degree polynomials. The testing results from the linear regression model and the second degree polynomial with 20 design variables are almost equally accurate. On the other hand, the double regression model is the best among the polynomial models. The double regression model is less accurate than the neural network in testing /fitting. However, the accuracy of the neural network in testing deteriorates more than the accuracy of the double regression model. Actually, the two tools have almost the same generalization performance. We selected the neural network translator has 22 design variables (Fig. 5.22a). The double regression model with 24 design variables was selected as the polynomial translator. 141 5.5.2 F/A Stiffness Figure 5.22b and Table 5.10 compare the results for the F/A. Similar trends as those for the I/O stiffness are observed: the neural network and the double regression are more accurate than either the second degree polynomials or the linear polynomial. The results of the neural network and the double regression model are equally accurate. For F/A stiffness, the neural network with 24 design variables was chosen because it had the smallest standard deviation for testing. The double regression model, which also has 24 design variables, had the smallest standard deviation for testing among all the polynomials, and was selected as the polynomial translator for the F/A stiffness. 5.5.3 Torsion Stiffness Results for torsion stiffness are slightly different than those for I/O and F/A stiffness. Figure 5.22c and Table 5.11 show that the double regression model is the most accurate among all models. The neural network and the second degree polynomials are almost equally accurate. The linear regression model is less accurate than the double regression model, but is slightly better than the neural networks or the second degree polynomials. The reason should be that the linear model has more design variables than the neural network and the second degree polynomial. Based on the above observations, the double regression model with 28 design variables was selected as the polynomial translator. The neural network with 20 design variables had the smallest standard deviation for testing, and was chosen as the neural network translator. 142 5.5.4 Mass The predictions of the mass are significantly more accurate compared with those for the I/O, F/A, and torsion stiffnesses (Fig. 5.22d, and Table 5.12). The standard deviations of the predictions from both neural networks and polynomials are less than 2.5 percent. The neural network is more accurate than the second degree polynomial. The double regression model is slightly better than the linear polynomial model. From the comparison, it is found that the double regression model, which has 31 design variables, gives the best prediction compared with the linear regression model or the second degree polynomial model. Therefore, the double regression model was selected as the polynomial translator. The neural network model with 24 design variables has the smallest standard deviation for testing compared with other neural networks. Therefore, it was selected as the neural network translator for the mass. 5.5.5 Validation We checked the above polynomial regression models using two criteria, namely, the CP criterion and AIC (Akaike’s Information Criterion). Figures 5.23a to 5.23d show the results of validating the regression models using CP criterion. Figures 5.24a-5.24d show the results when AIC criterion was used to validate the previous regression models. CP Criterion As Chapter 2 mentioned, the best model has a CP value almost equal to the number of terms of the polynomial. A value of CP greater than the number of terms generally indicates that the polynomial does not have enough terms or too many terms. For the I/O stiffness, the CP criterion is used to check the linear regression model and the second degree polynomial model with 20 design variables. The linear and the second degree polynomial models have 25 and 37 parameters (terms), respectively. Figure 5.23a shows that the CP value of the regression model is very close to the number of terms in the polynomial, p. Therefore, according to CP criterion, both regression models are appropriate. 143 Figure 5.23b shows the value of CP for the linear polynomial model and the second degree polynomial model with 21 design variables for the F/A stiffness. The two models have 25 and 37 parameters (terms), respectively. It is observed that both models are appropriate according to the CP criterion. Figure 5.23c shows the results for the torsion stiffness. It is found that both the linear polynomial model and the second degree polynomial model with 20 design variables are appropriate according to the CP criterion. Figure 5.23d shows the results for the mass using CP criterion. It is observed that the CP values for both the linear polynomial model and the second degree model with 20 design variables are very close to the number of parameters in the polynomials. If 36 parameters were used in the nonlinear regression model instead of 38, the value of CP would be closer to p. According to CP Criterion, and the regression model with 36 parameters probably would give better prediction than the translator with 38 parameters. AIC Criterion Figure 5.24a shows the AIC values of the linear and second degree polynomial models for the I/O stiffness. It is observed that the second degree polynomial with 20 design variables (37 parameters in the polynomial) has lower AIC value. According to AIC, the second degree polynomial model should have better generalization performance than the linear polynomial model. However, Table 5.9 shows that the linear polynomial is slightly more accurate than the second degree polynomial. Figure 5.24b shows the AIC values for the two polynomial models for the F/A stiffness, which have 25 and 37 parameters, respectively. It is observed that the second degree polynomial model has lower AIC value compared with the linear regression model. Therefore, the second degree polynomial should have better generalization 144 performance than the linear polynomial. Table 5.10 shows that the two models are almost equally accurate. Figure 5.24c shows the results for the torsion stiffness of A-pillar to roof rail joint. It is observed that the linear regression model and the second degree polynomial model with 20 design variables have almost the equal AIC value. Therefore, the two models should have same accurate generalization performance. Table 5.11 shows that the second degree polynomial is more accurate than the linear polynomial, but the difference of the two models is very small. Figure 5.23d shows the results for the mass of the A-pillar to roof rail joint. The linear polynomial model and the second polynomial model with 20 design variables have almost same AIC value. Although the second degree polynomial is slightly more accurate, both regression models will have almost same accurate generalization performance according to AIC criterion. The same conclusion can be drawn from Table 5.12. Discussion About the AIC Criterion It is found that the conclusions obtained using AIC criterion generally agree well with those obtained by using standard deviation of the bias corresponding to the testing set of the database. However, there are cases where the AIC criterion gives different conclusion from that obtained using standard deviation. For example, the fitting results of the linear regression model for I/O stiffness are worse than the second degree polynomial model. However, its testing results are better than the second degree polynomial model. The second degree polynomial model has smaller AIC value comparing with the linear model. According to the AIC criterion, the second degree polynomial model should give better prediction. The following are possible reasons for the conflicts. First, the linear and second degree models have almost the same accuracy (Tables 5.10-5.12) so even small numerical errors or statistical errors can affect the conclusions. Second, when we derive 145 the formulation of AIC, some assumptions are made, such as the distribution of each design variable, which may not be accurate. 5.5.6 Conclusions In this chapter, translators A for the A-pillar to roof rail joint were developed. Several regression models were studied and compared. CP and AIC criteria were used to validate obtained results. From the comparison, it was found that the double regression models were more accurate than the other polynomial models. Neural networks were also used to simulate the mapping between the design variables and the stiffness and mass of the joint. The double regression model and the neural network model seemed to have the same generalization performance. 146 Table 5.1: Effects of Mesh Size on FEA Results Seq. No 2 3 4 No. of Nodes 2098 3045 3635 No. of Elements 2079 3017 3537 I/O ( × 10 7 Nmm) 6.2716 6.2592 6.1042 F/A ( × 10 7 Nmm) 7.1630 7.0685 7.0145 Torsion ( × 10 7 Nmm) 1.1847 1.1851 1.1673 Mass (kg) 2.0302 2.0304 2.0297 Table 5.2: Comparison of FEA Results and Experimental Results Cars Stiffness/Mass I/O ( × 10 ) F/A( × 10 7 ) Torsion ( × 10 7 ) Mass I/O ( × 10 7 ) F/A ( × 10 7 ) Torsion ( × 10 7 ) Mass 7 Experimental (Nmm, kg) 6.531 4.929 1.259 3.399 5.981 1.044 FEA (Nmm, kg) 6.2716 7.1630 1.1847 2.0302 3.6003 5.0370 0.8449 2.4396 error -4.0 % 45.3% -5.9% 5.9% -15.8% -19.1% Car A Car B 147 Table 5.3: Measured Dimensions for A-pillar to Roof Rail Joints Cars and Dimensions Part1_part2_assmble Part2_part3_assmble Part3_part4_assmble Part5_part1_assmble Part6_part4_assmble Part7_part2_assmble Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Same_flange_width Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad Rr_width Rr_height Rr_h1 Rr_h2 Rr_angle_bottom Rr_blending_rad Ap_height Ap_inboard_width Ap_inboard_depth Ap_door_ws_distance Ap_window_depth Ap_flange1_angle Ap_reinf_depth Ap_reinf_flange1_angle Ap_reinf_inner_door_allowan ce Ap_reinf_flange2_angle Ap_blending_rad Part2_cut_distance Car 1 Car 2 Car 3 Car 4 Car 5 Car 6 Min Max Yes Yes Yes No No Yes 10 202 165 163 23 76 15 NO 20 15 16 35 90 90 90 115 100 38 30 80 25 16 95 65 55 40 35 35 90 60 26 25 35 22 0 9 155 38 91 55 90 Yes Yes Yes No No Yes 8 255 196 179 25 76 37 NO 24 16 17 45 103 90 130 112 90 62 23 90 19 15 140 73 75 52 54 30 88 75 31 27 19 26 10 9 140 29 130 100 55 Yes Yes Yes No Yes Yes 10 176 105 225 20 77 75 NO 18 16 19 35 90 102 110 96 87 27 46 78 30 13 111 56 47 39 51 24 60 52 20 36 24 0 Yes Yes Yes Yes No No 8 238 158 207 20 81 78 NO 20 14 12 50 90 110 90 122 90 63 14 108 34 15 146 83 77 60 53 42 53 78 23 29 34 20 Yes Yes Yes No No Yes 7 190 137 203 30 71 72 NO 24 16 14 33 90 100 83 123 98 54 20 109 28 17 159 84 63 60 60 35 108 60 23 36 23 19 7 Yes Yes Yes No No No 8 285 147 240 21 76 53 NO 19 16 16 40 113 104 110 107 63 8 96 28 12 210 146 79 108 107 20 105 66 33 36 21 17 6 7 176 105 163 20 71 15 18 14 12 33 90 90 90 96 87 27 8 78 19 12 96 56 47 39 35 20 53 52 20 25 19 17 0 9 140 22 91 55 55 10 285 196 240 30 81 78 24 16 19 50 113 110 130 123 107 63 46 109 34 17 210 146 79 108 107 42 108 78 33 36 35 26 10 9 155 38 130 138 110 22 116 105 85 120 96 113 110 138 148 Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 50 90 90 0.94 1.02 1.65 1.02 1.24 45 50 90 0.89 0.89 1.12 0.89 1.83 24 46 1.19 0.89 1.68 0.71 1.30 36 56 68 0.89 1.02 0.89 1.02 0.81 1.14 1.40 1.07 0.94 1.14 1.91 1.17 24 46 90 0.81 0.89 0.89 0.89 1.17 56 90 90 1.19 1.14 1.68 1.07 1.83 149 Table 5.4: Ranges of Design Variables for A-pillar to Roof Rail Joint No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Cars and Dimensions Distance_from_edge # Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width # Flange2_width # Flange3_width # Spot_weld_spacing # Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad Rr_width Rr_height Rr_h1 Rr_h2 Rr_angle_bottom Rr_blending_rad Ap_height Ap_inboard_width Ap_inboard_depth Ap_door_ws_distance Ap_window_depth Ap_flange1_angle Ap_reinf_depth Ap_reinf_flange1_angle Ap_reinf_inner_door_allowance Ap_reinf_flange2_angle Bounds (mm, degree) Lower Upper 8 8 160 314 95 216 148 264 18 33 65 89 14 86 20 20 16 16 16 16 50 50 90 124 90 121 90 143 90 135 90 118 25 69 7 51 71 120 17 37 11 19 87 231 51 161 43 87 35 119 32 118 18 46 48 119 47 86 18 36 23 40 17 39 15 29 0 11 1 29 91 171 20 69 91 144 150 39 Ap_blending_rad 40 Part2_cut_distance 41 Part3_cut_distance 42 Part4_extension 43 Part7_cut_distance 44 Thickness of Part1 45 Thickness of Part2 46 Thickness of Part3 47 Thickness of Part4 48 Thickness of Part7 # Fixed 50 50 22 42 40 0.74 0.81 0.81 0.81 1.06 151 121 62 99 121 1.31 1.25 1.85 1.18 2.01 151 Table 5.5: Ranking of Important Design Variables for the I/O Stiffness of A-pillar to Roof Rail Joint Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Design Variable A_pillar_offset AP_blending_rad Thickness of part3 AP_inboard_width Thickness of part7 AP_flange1_angle AP_door_ws_distance Phi Flange1_angle_up Flange1_angle_down AP_reinf_depth Theta Thickness of part2 AP_reinf_flange2_angle Angleback Header_horizontal_offset AP_window_depth Roof_rail_offset Flange2_angle_up RR_height 152 Table 5.6: Ranking of Important Design Variables for the F/A Stiffness of A-pillar to Roof Rail Joint Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Design Variable Thickness of part3 A_pillar_offset RR_height AP_height Theta Roof_rail_offset Thickness of part2 AP_door_ws_distance Thickness of part7 Flange1_angle_up Flange1_angle_down Flange2_fangle_up AP_reinf_inner_door_allowance AP_reinf_depth Door_allowance Phi AP_inboard_width RR_width AP_inboard_depth AP_window_depth 153 Table 5.7: Ranking of Important Design Variables for the Torsion Stiffness of A-pillar to Roof Rail Joint Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Design Variable A_pillar_offset Thickness of part3 Flange1_angle_down AP_inboard_width AP_blending_rad AP_door_ws_distance AP_window_depth AP_inboard_depth Roof_rail_offset Phi AP_flange1_angle Thickness of part2 Door_allowance Thickness of part7 RR_width RR_h1 AP_reinf_depth Theta Header_horizontal_offset RR_height 154 Table 5.8: Ranking of Important Design Variables for the Mass of A-pillar to Roof Rail Joint Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Design Variable Roof_rail_offset Thickness of part3 Header_offset Thickness of part7 A_pillar_offset Thickness of part1 RR_width Thickness of part2 Door_allowance H_width AP_blending_rad Part7_cut_distance Angleback Header_horizontal_offset Part4_extension AP_door_ws_distance H_blending_rad Part2_cut_distance AP_height AP_reinf_depth 155 Table 5.9: Comparison of Results from Different Methods For the I/O Stiffness of A-pillar to Roof Rail Joint Method Linear 2nd Degree Polynomial Double Regression (Cubic) Neural Networks Std of Fitting/Training 0.1583 0.1366 0.1231 0.1087 R2 83.5% 87.7% 88.1% Std of Testing 0.1625 0.1717 0.1210 0.1208 Table 5.10: Comparison of Results from Different Methods For the F/A Stiffness of A-pillar to Roof Rail Joint Method Linear 2nd Degree Polynomial Double Regression (Cubic) Neural Networks Std of Fitting/Training 0.1233 0.1056 0.0960 0.0920 R2 88.2% 90.5% 91.5% Std of Testing 0.1320 0.1312 0.1061 0.1011 Table 5.11: Comparison of Results from Different Methods For the Torsion Stiffness of A-pillar to Roof Rail Joint Method Linear 2nd Degree Polynomial Double Regression (Quadratic) Neural Networks Std of Fitting/Training 0.1391 0.1380 0.1229 0.1332 R2 82.6% 82.4% 84.7% Std of Testing 0.1488 0.1568 0.1362 0.1517 Table 5.12: Comparison of Results from Different Methods For the Mass of A-pillar to Roof Rail Joint Method Linear 2nd Degree Polynomial Double Regression (Quadratic) Neural Networks Std of Fitting/Training 0.0215 0.0197 0.0185 0.0155 R2 97.9% 98.2% 98.3% Std of Testing 0.0228 0.0242 0.0211 0.0219 156 Header Top Roof Rail Front A-pillar Figure 5.1a: A-pillar to Roof Rail Joint 157 Fixed Header Top Roof Rail Front Fixed I/O Stiffness FA Stiffness Torsion Stiffness Figure 5.1b: Definition of Stiffness for A-pillar to Roof Rail Joint 158 Part 1 Header Top Front Part 4 Roof Rail Part 3 Part 2 A-pillar Part 7 Figure 5.2: Parts of A-pillar to Roof Rail Joint 159 Part 6 Part 5 Figure 5.3: Parts of A-pillar to Roof Rail Joint 160 z Front of Car Origin Header x Inboard y z y x z O x x y z Origin Roof Rail y Origin A-pillar Figure 5.4: Global and Local Coordinate Systems 161 Header Header_vertical_offset y Header_horizontal_offset Roof_rail_offset z x z Roof Rail x Projection of Theta z y A-pillar y x Header_offset x A_pillar_offset Side View Projection of Phi x y x A-pillar_offset y Top View Figure 5.5: Orientation and Position of Each Branch 162 Header_offset H_blending_rad Theta Phi O RR_blending_rad AP_blending_rad Roof_rail_offset A_pillar_offset Figure 5.6: A-pillar to Roof Rail Joint Parameters 163 Flange1_width Flange3_width Distance_from_edge Flange2_width Spot_weld_Spacing Flange_width Flange_width can be Flange1_width, Flange2_width or Flange3_width Figure 5.7: Flange and Welding Dimensions 164 Up Front Flange3_width H_width z Flange1_width H_window_depth Flange3_angle_down H_height Flange1_angle_up y Flange1_angle_down Figure 5.8: Physical Parameters of Header 165 RR_width RR_h1 Angleback Flange3_angle_down RR_height Door_allowance z Flange2_angle_up y Flange2_angle_down Up Outboard RR_angle_bottom RR_h2 Figure 5.9: Physical Parameters of Roof Rail 166 AP_door_ws_distance Angleback AP_window_depth Flange1_angle_up AP_flange1_angle Flange1_angle_down AP_height AP_inboard_depth Door_allowance Outboard z y Flange2_angle_up Flange2_angle_down AP_inboard_width Figure 5.10: Physical Parameters of A-pillar 167 AP_reinf_flange1_angle Ap_reinf_depth AP_reinf_inner_ door_allowance Outboard AP_reinf_flange2_angle Figure 5.11: Physical Parameters for A-pillar Reinforcement 168 Part2_cut_distance Header_offset Figure 5.12: Physical Parameters for Part 2 169 Part3_cut_distance Header_offset Figure 5.13: Physical Parameters for Part 3 170 Part4_extension Figure 5.14: Physical Parameter for Part 4 171 Part7_cut_distance Header_offset Figure 5.15: Physical Parameters for Part 7 172 Y Z X Figure 5.16: FEA Model for A-pillar to Roof Rail Joint 173 Relation Between Torsional Stiffness (FEA) and the Values of (Thickness of Part3 / A_pillar_Offset) for A_pillar to Roof Rail Joint 4.0E7 3.5E7 3.0E7 Torsional Stiffness (FEA) 2.5E7 2.0E7 1.5E7 1.0E7 5.0E6 0.0E0 0.002 0.004 0.006 0.008 0.010 0.012 Thickness of Part3 / A_pillar_Offset Fig. 5.17: Correlation Between Torsion Stiffness and the Value of (Thickness of Part3/A_pillar_Offset) for A-pillar to Roof Rail Joint 174 1.8E8 1.6E8 1.4E8 Explanation of Separating Designs into Two Regression Model Comparison Results the FEA Results and the Prediction Comparison of FEA Betweenand Predictions from Linear Groups Comparison Between FEA Model for to Roof Rail Joint From thefor the I/O StiffnessandA-pillar A-pillar to Roof Rail Joint (I/O) Linear Regression of Double Regression Results (I/O) for the I/O Stiffnessof A-pillar to Roof Rail Joint 2.0E8 Comparison of FEA Results and Predictions from Linear Regression Model for the F/A Stiffness of A-pillar to Roof Rail Joint 1.5E8 FEA Results FEA Results FEA Results 1.2E8 1.0E8 8.0E7 6.0E7 4.0E7 2.0E7 0.0E0 0.0E0 FEA Results 2.0E7 4.0E7 2.0E7 6.0E7 4.0E7 8.0E7 1.0E8 6.0E7 1.2E8 8.0E7 1.4E8 1.6E8 1.0E8 1.8E8 1.2E8 1.0E8 5.0E7 Prediction From Linear Double Regression Predicted I/O Stiffness Prediction from Linear Regression Model Predicted I/O Stiffness fromRegression Model 0.0E0 0.0E0 5.0E7 1.0E8 1.5E8 Prediction from Linear Regression Model 4.0E7 Comparison of FEA Results and Predictions from Linear Regression Mode for the Torsion Stiffness of A-pillar to Roof Rail Joint 5.0 Comparison of FEA Results and Predictions from Linear Regression Model for the Mass of A-pillar to Roof Rail Joint 3.5E7 4.5 3.0E7 4.0 2.5E7 FEA Results FEA Results 1.0E7 2.0E7 3.0E7 3.5 2.0E7 1.5E7 3.0 1.0E7 2.5 5.0E6 2.0 0.0E0 0.0E0 Prediction from Linear Regression Model 1.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Prediction from Linear Regression Model Figure 5.18:. Comparison of FEA Results and Predictions from Linear Regression Model 175 Comparison of FEA Results and Predictions from Quadratic Model for the I/O Stiffness of A-pillar to Roof Rail Joint 1.8E8 1.6E8 2.0E8 1.8E8 1.6E8 Comparison of FEA Results and Predictions from Quadratic Model for the F/A Stiffness of A-pillar to Roof Rail Joint 1.4E8 1.4E8 1.2E8 FEA Results 1.0E8 8.0E7 6.0E7 4.0E7 2.0E7 0.0E0 0.0E0 FEA Results 1.2E8 1.0E8 8.0E7 6.0E7 4.0E7 2.0E7 0.0E0 0.0E0 2.0E7 4.0E7 6.0E7 8.0E7 1.0E8 1.2E8 1.4E8 5.0E7 1.0E8 1.5E8 Prediction from Quadratic Regression Model Prediction from Quadratic Regression Model Comparison of FEA Results and Predictions from Quadratic Model 4.0E7 for the Torsion Stiffness of A-pillar to Roof Rail Joint 5.0 Comparison of FEA Results and Predictions from Quadratic Model for the Mass of A-pillar to Roof Rail Joint 3.5E7 4.5 3.0E7 4.0 2.5E7 FEA Results 5.0E6 1.0E7 1.5E7 2.0E7 2.5E7 3.0E7 FEA Results 3.5 2.0E7 1.5E7 3.0 1.0E7 2.5 5.0E6 2.0 0.0E0 0.0E0 1.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Prediction from Quadratic Regression Model Prediction from Quadratic Regression Model Figure 5.19: Comparison of FEA Results and Predictions from Second Degree Polynomial Model 176 3.0E7 Relation Between the FEA Results and the Error of Prediction for the Linear Regression Model of I/O Stiffness 2.0E7 (Prediction - FEA Results) 1.0E7 0.0E0 -1.0E7 -2.0E7 -3.0E7 0.0E0 2.0E7 4.0E7 6.0E7 8.0E7 1.0E8 1.2E8 1.4E8 FEA Results Fig 5.20a: Relation Between FEA Results and Error of Prediction for the Linear Polynomial Model of I/O Stiffness 177 Comparison Between the FEA Results and the Prediction 1.8E8 1.6E8 1.4E8 1.2E8 1.0E8 8.0E7 6.0E7 4.0E7 2.0E7 0.0E0 0.0E0 From the Linear Regression Model for A-pillar to Roof Rail Joint (I/O) FEA Results 2.0E7 4.0E7 6.0E7 8.0E7 1.0E8 1.2E8 Predicted I/O Stiffness Figure 5.20b: Comparison of FEA Results and Prediction from Linear Polynomial Model and Explanation of Double Regression 178 C o m p a ris o n o f F E A R e su lts an d F ittin g R e su lts fro m D o u b le R egres sio n fo r th e I/O S tiffn es s o f A -p illa r to R o o f R ail J o in t 1.8E + 08 1.8E+ 08 C o m p a r is o n o f F E A R e s u lts a n d F ittin g R e s u lts fr o m D o u b le R e gr e s s io n f o r th e F /A S tif fn e s s o f A - p illa r to R o o f R a il J o in t 1.6E + 08 1.6E+ 08 1.4E + 08 1.4E+ 08 1.2E + 08 1.2E+ 08 F E A R e su lts F E A R e s u lts 1.0E + 08 1.0E+ 08 8.0E + 07 8.0E+ 07 6.0E + 07 6.0E+ 07 4.0E + 07 4.0E+ 07 2.0E + 07 2.0E+ 07 0.0E + 00 0.0E+ 00 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 R e su lts F ro m D o u b le R egre ssio n R e s u lts F r o m D o u b le R e g r e s s io n C o m p ariso n o f F E A R e su lts a n d F ittin g R es u lts fro m D o u b le R egre ssio n fo r th e T o rs io n S tiffn e ss o f A -p illa r to R o o f R a il Jo in t 4.0E +0 7 4 .5 C o m p a riso n o f F E A R e su lts a n d F ittin g R e su lts fro m D o u b le R e gre ss io n fo r th e M a ss o f A -p illar to R o o f R a il Jo in t 3.5E +0 7 4 .0 3.0E +0 7 3 .5 2.5E +0 7 F E A R e su lts 2.0E +0 7 F E A R e su lts 3 .0 1.5E +0 7 2 .5 1.0E +0 7 2 .0 5.0E +0 6 0.0E +0 0 1 .5 0 .0 E + 0 0 5 .0 E + 0 6 1 .0 E + 0 7 1 .5 E + 0 7 2 .0 E + 0 7 2 .5 E + 0 7 3 .0 E + 0 7 3 .5 E + 0 7 4 .0 E + 0 7 1.5 2.0 2.5 3.0 3.5 4.0 4.5 R es u lts F ro m D o u b le R egre ssio n R e sults F ro m D o ub le R e gre ssio n Figure 5.21a: Scatter Plots for the Fitting Results of Polynomial Translators 179 C o m p a ris o n o f F E A R e su lts an d T e stin g R e su lts fro m D o u b le R e gre ss io n fo r th e I/O S tiffn es s o f A -p illa r to R o o f R ail J o in t 1.8E + 08 1.8E+ 08 C o m p a r is o n o f F E A R e s u lts a n d T e s tin g R e s u lts f ro m D o u b le R e gr e s s io n f o r th e F /A S tif fn e s s o f A - p illa r to R o o f R a il J o in t 1.6E + 08 1.6E+ 08 1.4E + 08 1.4E+ 08 1.2E + 08 1.2E+ 08 F E A R e su lts F E A R e s u lts 1.0E + 08 1.0E+ 08 8.0E + 07 8.0E+ 07 6.0E + 07 6.0E+ 07 4.0E + 07 4.0E+ 07 2.0E + 07 2.0E+ 07 0.0E + 00 0.0E+ 00 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 R e su lts F ro m D o u b le R egre ssio n R e s u lts F r o m D o u b le R e g r e s s io n C o m p ariso n o f F E A R e su lts a n d T e stin g R e su lts fro m D o u b le R e gres sio n fo r th e T o rs io n S tiffn e ss o f A -p illa r to R o o f R a il Jo in t 4.0E +0 7 4 .5 C o m p a riso n o f F E A R e su lts a n d T es tin g R es u lts fro m D o u b le R e gre ssio n fo r th e M a ss o f A -p illar to R o o f R a il Jo in t 3.5E +0 7 4 .0 3.0E +0 7 3 .5 2.5E +0 7 F E A R e su lts 2.0E +0 7 F E A R e su lts 3 .0 1.5E +0 7 2 .5 1.0E +0 7 2 .0 5.0E +0 6 0.0E +0 0 1 .5 0 .0 E + 0 0 5 .0 E + 0 6 1 .0 E + 0 7 1 .5 E + 0 7 2 .0 E + 0 7 2 .5 E + 0 7 3 .0 E + 0 7 3 .5 E + 0 7 4 .0 E + 0 7 1.5 2.0 2.5 3.0 3.5 4.0 4.5 R es u lts F ro m D o u b le R egre ssio n R e sults F ro m D o ub le R e gre ssio n Figure 5.21b: Scatter Plots for the Testing Results of Polynomial Translators 180 C o m p a ris o n o f F E A R e su lts an d N N T ra in in g R e su lts fo r th e I/O S tiffn es s o f A -p illa r to R o o f R ail J o in t 1.8E + 08 1.8E+ 08 C o m p a ris o n o f F E A R e s u lts a n d N N T r a in in g R e s u lts f o r th e F /A S tiff n e s s o f A -p illa r to R o o f R a il J o in t 1.6E + 08 1.6E+ 08 1.4E + 08 1.4E+ 08 1.2E + 08 1.2E+ 08 F E A R e su lts F E A R e s u lts 1.0E + 08 1.0E+ 08 8.0E + 07 8.0E+ 07 6.0E + 07 6.0E+ 07 4.0E + 07 4.0E+ 07 2.0E + 07 2.0E+ 07 0.0E + 00 0.0E+ 00 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 N N T ra in in g R e su lts N N Tr ain in g R e s u lts C o m p ariso n o f F E A R e su lts a n d N N T ra in in g R e su lts fo r th e T o rs io n S tiffn e ss o f A -p illa r to R o o f R a il Jo in t 4.0E +0 7 4 .5 C o m p a ris o n o f F E A R e su lts a n d N N T ra in in g R e su lts fo r th e M as s o f A -p illa r to R o o f R a il Jo in t 3.5E +0 7 4 .0 3.0E +0 7 3 .5 2.5E +0 7 F E A R e su lts 2.0E +0 7 F E A R e su lts 3 .0 1.5E +0 7 2 .5 1.0E +0 7 2 .0 5.0E +0 6 0.0E +0 0 1 .5 0 .0 E + 0 0 5 .0 E + 0 6 1 .0 E + 0 7 1 .5 E + 0 7 2 .0 E + 0 7 2 .5 E + 0 7 3 .0 E + 0 7 3 .5 E + 0 7 4 .0 E + 0 7 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N N T ra in in g R es u lts N N T ra ining R e sults Figure 5.21c: Scatter Plots for the Training Results of Neural Network Translators 181 C o m p a riso n o f F E A R esu lts a n d N N T estin g R e su lts fo r th e I/O S tiffn e ss o f A -p illa r to R o o f R ail Jo in t 1.8E + 08 1.8E+ 08 C o m p a ris o n o f F E A R e s u lts a n d N N T e s tin g R e s u lts f o r th e F /A S tiff n e s s o f A -p illa r to R o o f R a il J o in t 1.6E + 08 1.6E+ 08 1.4E + 08 1.4E+ 08 1.2E + 08 1.2E+ 08 F E A R e su lts F E A R e s u lts 1.0E + 08 1.0E+ 08 8.0E + 07 8.0E+ 07 6.0E + 07 6.0E+ 07 4.0E + 07 4.0E+ 07 2.0E + 07 2.0E+ 07 0.0E + 00 0.0E+ 00 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 1 .2 E + 0 8 1 .4 E + 0 8 1 .6 E + 0 8 1 .8 E + 0 8 N N T estin g R es u lts N N Te s tin g R e s u lts C o m p ariso n o f F E A R e su lts a n d N N T es tin g R e su lts fo r th e T o rs io n S tiffn e ss o f A -p illa r to R o o f R a il Jo in t 4.0E +0 7 4 .5 C o m p a ris o n o f F E A R e su lts a n d N N T e stin g R es u lts fo r th e M as s o f A -p illa r to R o o f R a il Jo in t 3.5E +0 7 4 .0 3.0E +0 7 3 .5 2.5E +0 7 F E A R e su lts 2.0E +0 7 F E A R e su lts 3 .0 1.5E +0 7 2 .5 1.0E +0 7 2 .0 5.0E +0 6 0.0E +0 0 1 .5 0 .0 E + 0 0 5 .0 E + 0 6 1 .0 E + 0 7 1 .5 E + 0 7 2 .0 E + 0 7 2 .5 E + 0 7 3 .0 E + 0 7 3 .5 E + 0 7 4 .0 E + 0 7 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N N T e stin g R e su lts N N T e sting R e sults Figure 5.21d: Scatter Plots for the Testing Results of Neural Network Results 182 Results for the I/O Stiffness of A-pillar to Roof Rail Joint 0.22 0.20 0.18 Fitting (Linear) Testing (Linear) Fitting (Double Regression) Testing (Double Regression) Standard Deviation 0.16 0.14 0.12 0.10 Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing 13 14 15 16 17 18 19 20 21 22 23 24 25 0.08 12 Number of Design Variables Fig. 5.22a: Comparison of Results for I/O Stiffness of A-pillar to Roof Rail Joint 183 Results for the F/A Stiffness of A-pillar to Roof Rail Joint 0.22 0.21 0.20 0.19 0.18 0.17 Standard Deviation 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 12 13 14 15 16 17 18 Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing Fitting (Linear) Testing (Linear) Fitting (Double Regression) Testing (Double Regression) 19 20 21 22 23 24 25 Number of Design Variables Figure 5.22b: Comparison of Results for F/A Stiffness of A-pillar to Roof Rail Joint 184 Results for Torsion Stiffness of A-pillar to Roof Rail Joint 0.22 Neural Network Training Neural Network Testing 0.2 Polynomail Fitting Polynomial Testing Fitting (Linear) Testing (Linear) Fitting (Double Regression) Testing (Double Regressin) 0.18 Standard Deviation 0.16 0.14 0.12 0.1 12 14 16 18 20 22 24 Number of Design Variables 26 28 30 Figure 5.22c: Comparison of Results for Torsion Stiffness of A-pillar to Roof Rail Joint 185 Results fo the Mass of A-pillar to Roof Rail Joint 0.04 Neural Network Training Neural Network Testing Polynomial Fitting Polynomial Testing Fitting (Linear) Testing (Linear) Fitting (Double Regression) Testing (Double Regression) 0.035 Standard Deviation 0.03 0.025 0.02 0.015 0.01 12 14 16 18 20 22 24 Number of Design Variables 26 28 30 32 Figure 5.22d: Comparison of Results for the Mass of A-pillar to Roof Rail Joint 186 200 Relation p for the Cp and p Relation Between Cp and Between Nonlinear Model of I/O Stiffness for of A-pillar to Roofof I/OJoint (use of A-pillar to Roof Rail Joint the Linear Model Rail Stiffness 20 Design Variables) 150 Cp Cp Cp Cp 100 50 0 0 2 4 6 58 10 12 14 16 18 20 22 24 26 28 20 32 34 3625 10 15 30 p (Number of Parameters in Polynomial) p (Number of Parameters in Polynomial) 200 Relation p for the Cp and p Relation Between Cp and Between Nonlinear Model of I/O Stiffness for of A-pillar to Roofof I/OJoint (use of A-pillar to Roof Rail Joint the Linear Model Rail Stiffness 20 Design Variables) 150 Cp Cp Cp Cp 100 50 0 0 2 4 6 58 10 12 14 16 18 20 22 24 26 28 20 32 34 3625 10 15 30 p (Number of Parameters in Polynomial) p (Number of Parameters in Polynomial) Figure 5.23a: Validating Using Cp Criterion for I/O Stiffness 187 200 Relation Between Cp and p for the Linear Model of F/A Stiffness of A-pillar to Roof Rail Joint 150 Cp Cp 100 50 0 0 5 10 15 20 25 p (Number of Parameters in Polynomial) 200 Relation Between Cp and p for the p and p Relation Between CNonlinear Model of F/A Stiffness forof A-pillar to Roof of F/A Stiffness21 Design Variables) Joint the Linear Model Rail Joint (use of A-pillar to Roof Rail Cp 150 Cp Cp Cp 100 50 0 0 2 4 6 58 10 12 14 16 18 20 22 24 26 28 20 32 34 3625 10 15 30 p p (Number of Parameters in Polynomial) (Number of Parameters in Polynomial) Figure 5.23b: Validating Using Cp Criterion for F/A Stiffness 188 Relation Between Cp and p 200 for the Linear Model of Torsion Stiffness of A-pillar to Roof Rail Joint 150 Cp Cp 100 50 0 0 5 10 15 20 25 p (Number of Parameters in Polynomial) Relation BetweenRelation Between C and p Model of Torsion Stiffness Cp and p for the Nonlinear p 200 for of A-pillar to Roofof Torsion (use 20 Design Variables) Rail Joint the Linear Model Rail Joint Stiffness of A-pillar to Roof 150 Cp Cp CCp p 100 50 0 0 55 10 10 15 15 20 20 25 25 p (Number of of Parameters in Polynomial) p (Number Parameters in Polynomial) Figure 5.23c: Validating Using Cp Criterion for Torsion Stiffness 189 200 Relation Between Cp and p for the Linear Model of Mass of A-pillar to Roof Rail Joint Cp 150 Cp 100 50 0 0 5 10 15 20 25 30 p (Number of Parameters in Polynomial) Relation Between Cp and p for the Nonlinear Model of Mass Relation Between Cp and p 200 of A-pillar to Roof Rail Joint (useA-pillar to Roof Rail Joint for the Linear Model of Mass of 20 Design Variables) 150 Cp Cp C Cpp 100 50 0 0 2 4 5 6 8 10 10 14 16 15 20 22 20 26 28 25 32 34 30 38 12 18 24 30 36 p (Number of Parameters in Polynomial) p (Number of Parameters in Polynomial) Figure 5.23d: Validating Using Cp Criterion for Mass 190 Relation Between AIC and p for I/O Stiffness of A-pillar to Roof Rail Joint 1.32E4 1.31E4 linear model nonlinear model using 20 design variables 1.30E4 AIC final point 1.29E4 final point 1.28E4 1.27E4 1.26E4 0 5 10 15 20 25 30 35 40 45 p (Number of Parameters in Polynomial) Figure 5.24a: Validating Using AIC Criterion for the I/O Stiffness of A-pillar to Roof Rail Joint 191 Relation Between AIC and p for F/A Stiffness of A-pillar to Roof Rail Joint 1.32E4 1.31E4 linear model 1.30E4 nonlinear model using 21 design variables AIC 1.29E4 final point final point 1.28E4 1.27E4 1.26E4 0 5 10 15 20 25 30 35 40 45 p (Number of Parameters in Polynomial) Figure 5.24b Validating Using AIC Criterion for the F/A Stiffness of A-pillar to Roof Rail Joint 192 Relation Between AIC and p for Torsion Stiffness of A-pillar to Roof Rail Joint 1.58E4 linear model 1.57E4 nonlinear model using 20 design variables AIC 1.56E4 final point 1.55E4 final point 1.54E4 0 5 10 15 20 25 30 35 p (Number of Parameters in Polynomial) Figure 5.24c Validating Using AIC Criterion for the Torsion Stiffness of A-pillar to Roof Rail Joint 193 Relation Between AIC and p for the Mass of A-pillar to Roof Rail Joint -1500 -1600 linear model nonlinear model using 20 design variables -1700 AIC -1800 -1900 final point -2000 final point -2100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 p (Number of Parameters in Polynomial) Figure 5.24d: Validating Using AIC Criterion for the Mass of A-pillar to Roof Rail Joint 194 Chapter 6 Developing Translator B for an Actual B-pillar to Rocker Joint 6.1 Introduction This chapter applies the general methodology for developing translator B in Chapter 3 to a B-pillar to rocker joint. The B-pillar to rocker joint, which lies between the front and rear doors, is a T- like joint (see Chapter 4 for a detailed description of this joint). The vertical branch of the joint is called B-pillar, and the horizontal branch is called rocker. This chapter presents work that was completed in close collaboration with Mr. Ling. Therefore, parts of this chapter appear in his thesis (Ling, 1998). This chapter is organized as follows: • Section 6.2 formulates the optimization problem used in developing translator B. This section explains the selection and classification of design variables, the objective function, and the constraints. • Section 6.3 presents some applications of translator B to some design problems. First, we use several methods to check the convergence of translator B. Then, we compare the optimum results from translator B with FEA results. We also compare the response surface polynomial (RSP) and neural network (NN) results since we use both to develop translator B. Then, we perform a parametric study to examine the effects of stiffness requirements on the mass of the optimum design, and the relation between objective function and the lower and upper bounds of some important design variables. Design guidelines for improving the design of the B-pillar to rocker joint are presented. The results are discussed and compared with those in other studies. 195 6.2 Formulation of the Optimization Problem for Developing a Translator B for a B-pillar to Rocker Joint 6.2.1 Definition of the Problem Translator B uses optimization to find the most efficient design that meets given performance target (stiffness requirements), and satisfies all the constraints. Unlike translator A, which calculates the response of a joint whose design variables are specified, translator B solves the inverse problem. The user specifies performance requirements for the joint, including, stiffness requirements and constraints on the design variables, and translator B finds the most efficient design that satisfies all these requirements. Structural optimization is an iterative process, and it would be very time-consuming if during the iteration, each new design were analyzed using FEA. To avoid this problem, a translator A is used to simulate FEA. As explained in Chapters 2 and 4, translator A is created using a continuous function to simulate the relation between the performance characteristics (stiffness and mass) and the values of the design variables of a design. Translator A predicts stiffness and mass almost instantaneously for a new design once its dimensions are specified. Thus, use of translator A enables the user to optimize the joint design. Two types of translator A are used in this study. One is created using response surface polynomials, and the other is created using neural networks. There are different types of construction for B-pillar to rocker joints. As explained in Chapter 4, different types of construction are primarily characterized by different reinforcements. Some reinforcements are more common than others. For example, an extended pillar reinforcement is common. After consulting engineers from an automotive company, we decided to consider only one type of construction (Figs.4.2 and 4.3a). This type of B-pillar to rocker joint has an extended pillar reinforcement, and the center plate is connected to the bottom of the front rocker and the rear plate of the rocker (This rocker section is called generic type of rocker cross section. Fig. 4.6). Bulkheads are not considered in the model. Ling (1998) studied the effects of bulkheads on the stiffness. 196 Pillar bridge is not considered in the model because it is not effective (Murphy, 1995). Translator B uses 48 design variables. 6.2.2 Design Variables The design variables for the B-pillar to rocker joint are shown in Table 6.1. As explained in Chapter 3, the design variables are divided into four types: 1) Design variables that are fixed by the optimization program because of design rules or conventions. An example is the length of rocker. 2) Design variables that are fixed by the user. 3) Dependent design variables whose values can be expressed in terms of other design variables. 4) Independent design variables that can change in optimization. As explained in Chapter 4, some design variables are generally fixed because of packaging, and manufacturing requirements. The values of some design variables are determined from styling requirements. To compare the stiffness of different joint designs fairly, the rocker branch and B-pillar branch should be cut at the same length for different joint designs. There is one design variable in the first type (fixed by the program), namely, rocker_length (Fig. 4.10). We fix it at 485 mm in order to compare fairly the stiffness of different joints. There are seven design variables in the second type (fixed by the user), namely, length_of_flange, spot_weld_spacing, spot_weld_placement, pillar_angle, pillar_io_angle, pillar_location, and aft_pillar_hole (Figs.4.13, 4.7, 4.10, and 4.14). Because of manufacturing considerations, length_of_flange, spot_weld_spacing, and spot_weld_placement are fixed at 19 mm, 47.5 mm, and 9.5 mm, respectively. Pillar_angle and pillar_io_angle are fixed because the designer generally fixes the orientation of the B-pillar. Pillar_location is fixed because of door packaging considerations. 197 There are six design variables whose values depend on other independent design variables, namely, pillar_height, fwd_outer_ver_blending_rad, fwd_outer_hor_blending_rad, aft_outer_ver_blending_rad, aft_outer_hor_blending_rad, and pillar_reinforcement_depth (Figs. 4.10, 4.14, and 4.11). The length of the B-pillar branch is defined as the distance from the center of rocker central line to the top of the Bpillar (Fig. 6.1). The length of the B-pillar branch is fixed at 269 mm to allow for fair comparison of the stiffness and mass of different joints. Pillar_height is a dependent variable, which can be expressed in terms of the length of B-pillar branch and other design variables. The blending radii at the outer side of B-pillar, such as fwd_outer_ver_blending_rad, are assumed equal to their corresponding blending radii at the inner side of B-pillar, such as fwd_inner_ver_blending_rad. In this study, pillar_reinforcement_depth is a dependent design variable because we assume that the pillar reinforcement touches the bottom of the rocker. Table 6.1 classifies the design variables. The optimization program does not directly change the design variables that are fixed by the program or by the user, and design variables that are dependent. Only 34 independent design variables are considered. The ranges of the design variables are shown in Table 6.1. 6.2.3 Objective Function The objective function of translator B for the B-pillar to rocker joint can be expressed as: ˆ ˆ ˆ F = αM + (1 − α ) (( K I / O − K I / O ) / K I / O ) 2 + (( K F / A − K F / A ) / K F / A ) 2 + (( K Tor − K Tor ) / K Tor ) 2 where α is a weighting factor. M is the mass of the joint. K I / O , K F / A , and K Tor ˆ ˆ ˆ I/O, F/A and torsion stiffnesses of the joint. K , K , and K are the user-specified I /O F/A Tor (6.1) are the requirements (targets) for I/O, F/A and torsion stiffness. K I / O , K F / A , and K Tor are the stiffness values used to normalize the stiffness. Generally, mass varies from 2 to 8 kg. Each stiffness term varies from 0 to 3. We normalize the stiffness so that the mass and the stiffnesses can have compatible magnitude. 198 In this study, α is set to be 1. The objective function is equal to the mass. However, α can be set to be any value between 0 and 1. In this case, the objective function will be the sum of mass and some measure of difference between the stiffness of a design and the given stiffness targets. The optimum design obtained using this objective function will be a joint that not only has low mass but also its stiffness is close to the stiffness requirement. 6.2.4 Constraints There are five types of constraints, namely, packaging, manufacturing, styling, mathematical, and performance target constraints. Packaging constraints are related to the arrangement of car components in space. Manufacturing constraints are due to manufacturing limitations. Styling constraints are due to styling requirements. Mathematical constraints are used to control the range of different design variables to ensure that a design has a feasible geometry. Performance constraints dictate that the stiffnesses should exceed given minimum values. The constraints used in translator B are explained in the following section. The number in the parenthesis of each constraint is the corresponding constraint number in the program for translator B. • Packaging Constraints 1) Space must be provided for the door latch (Fig. 4.9) (4). Pillar_outer_length - pillar_inner_length † 0 (6.2) 2) There should be enough room between the front and rear doors (Fig. 4.9)(52). Minimum value of pillar_outer_length - pillar_outer_length † 0 (6.3) 199 3) There is a minimum requirement for the thickness of door (Fig. 4.9)(53). Minimum value of outer_pillar_width - outer_pillar_width † 0 (6.4) 4-7) Orientation of B-pillar must be within a range so that it is easy to get in and out of the car (Fig.4.7)(54-57). Minimum value of pillar_angle - pillar_angle † 0 Pillar_angle - maximum value of pillar_angle † 0 Minimum value of pillar_io_angle - pillar_io_angle †0 Pillar_io_angle - maximum value of pillar_io_angle † 0 (6.5) (6.6) (6.7) (6.8) 8) Pillar reinforcement should not intersect with the bottom and/or and side of the rocker (Fig. 6.2)(5). Case 1: Pillar reinforcement intersects the front part of rocker Case 2: Pillar reinforcement intersects the bottom part of rocker (6.9b) (6.9a) 9-10) Pillar_reinforcement_depth must be within its range (Fig. 4.11) (84,85). Minimum value of pillar_reinforcement_depth Pillar_reinforcement_depth† 0 Pillar_reinforcement_depth – maximum value of pillar_reinforcement_depth †0 11-18) Blending radii should be within the range [Bld_Mi, Bld_Mx] (Fig. 4.10)(58-65) because of door packaging considerations. Bld_Mi and Bld_Mx are the minimum and maximum values given by the user. Their default values are 30mm and 200mm, respectively. (6.11) (6.10) 200 Bld_Mi -Fwd_inner_ver_blendign_rad †0 Fwd_inner_ver_blendign_rad - Bld_Mx †0 Bld_Mi - Aft_inner_ver_blendign_rad †0 Aft_inner_ver_blendign_rad - Bld_Mx †0 Bld_Mi - Fwd_inner_hor_blendign_rad †0 Fwd_inner_hor_blendign_rad - Bld_Mx †0 Bld_Mi - Aft_inner_hor_blendign_rad †0 Aft_inner_hor_blendign_rad - Bld_Mx †0 (6.12) (6.13) (6.14) (6.15) (6.16) (6.17) (6.18) (6.19) 19-20) Rocker height and width must not exceed a maximum value (Fig.4.8) (40,41). Rocker_height - maximum value of rocker_height †0 Rocker_width - maximum value of rocker_width †0 (6.20) (6.21) 21,22) The door edge height and width must be large enough to accommodate the sealant and the door edge (Fig. 4.8) (42,43). Minimum value of door_edge_height -door_edge_height †0 Minimum value of door_edge_width - door_edge_width † 0 (6.22) (6.23) 23,24) There is a range for A1 to allow for water drainage (Fig. 4.8) (44,45). Minimum value of A1-A1 † 0 A1-90 †0 25,26) There is a range for A5 from seat packaging considerations (Fig.4.8)(46,47). Minimum value of A5 - A5 † 0 A5 – maximum value of A5 † 0 (6.26) (6.27) (6.24) (6.25) 201 27,28) There is a range for A3 from packaging consideration (Fig. 4.8) (49,50). Minimum value of A3-A3†0 A3- maximum value of A3†0 29) There must be a slope for water to run off (Fig. 6.3a) (51). A6Mi (Minimum value of A6) – A6 † 0 (6.30) (6.28) (6.29) where A6Mi is given by the user. A default value of 3 degree was used in the examples of this chapter. 30) The slope of top plate of inner rocker should not be too high (Fig.6.5) (72). Atan(d7/h4) – SRMx (maximum value of the slope) †0 (6.31) The user can define SRMx. A default value of 10 degree was used in the following examples. 31) The width of inner rocker cell can not be too large because of seat packaging considerations (Fig. 4.8) (83). Rocker_width - outboard_cell_width – widInRkMx (maximum value) †0 (6.32) where widInRkMx is defined by the user. A default value of 82mm was used in the following examples. 202 32) There should be enough room to put afterward bulkhead (Figs. 4.10 an 4.14) (9). Aft_bulk_head_position + aft_inner_hor_blendign_rad pillar_location†0 (6.33) 33) There should be room to put forward bulkhead (Figs. 4.10 and 4.14)(6,7) Pillar_location + fwd_inner_hor_blendign_rad+pillar_base + pillar_inner_length-pillar_outer_length- fwd_bulk_head_position †0 (6.34) 34) Inner_pillar_base_width should be smaller than the horizontal projection of BR_L8 to connect the pillar back with the inner plate of rocker (Figs. 4.15 and 6.3) (71). Inner_pillar_base_width-horizontal distance of BR_L8†0 • Manufacturing Constraints Manufacturing constraints include stamping and welding constraints. One needs very detailed information about the geometry to determine if a given design is feasible. In the early design stages this information is not available. We have used crude equations to check if a design is manufacturable. Stamping constraints include constraints for strains, spring back, and die lock. To ensure that plastic strains are low, we impose low limits on the edges of each section, and the angles between two adjacent plates of a section. We also impose an upper limit on the depth of draw to avoid overstretching a plate (Fig.6.8a). Spring back constraints ensure that a plate can be permanently bent. For this reason, the deformation angle should exceed a minimum value (Fig.6.6). Constraints on die lock ensure that the die and block can be separated after the plate has been stamped. For this purpose, the vertical walls of the hat-shape section should be slanted away from the center of the section (Fig. 6.7a). (6.35) 203 35-42) The length of each edge on the rocker section should be greater than a minimum value, so that the rocker inner and outer shells can be stamped (Fig. 6.3) (12-19). RMin_L (minimum length) – BR_Lj † 0 j=1,…,8 (6.36) where RMin_L is minimum length. The user can define this length. A default value of 10 mm was used in the examples of this chapter. 43-50) The angle between two adjacent edges of the rocker section must be larger than a minimum value to avoid a sharp angle between the plates (Fig. 6.4) (20-27). RMin_A (minimum angle) – BR_Anglej † 0 j=1,…,8 (6.37) where RMin_A is minimum angle. The user can define this angle. A default value of 20 degrees was used in the examples of this chapter. 51,52) The top and bottom plates of the outer rocker shell should be able to avoid die lock (Figs. 4.8, 6.7a and 6.7b)(28,29). Die lock angle is the minimum requirement on the angle between the moving direction of a die and the side shell of a die casting part. This constraint is due to manufacturing consideration. ADieMi (minimum value of die angle) – (90-Aj) † 0 j=1,2 (6.38) where the minimum angle ADieMi is defined by the user. A default value of 3 degrees was used in the examples of this chapter. 204 53) The draw angle of the front rocker should be smaller than a maximum value from stamping considerations (Fig.6.6) (30). Draw angle is the maximum allowable angle between the moving direction of a die and normal direction of a shell metal from which a die casting part is made. Draw angle of front rocker– AdrawMx (maximum value)† 0 (6.39) where AdrawMx is a maximum angle. The user can define this angle. A default value of 30 degrees was used in the examples of this chapter. 54) The ratio of depth of draw over the width of the draw should be smaller than a given value to avoid excessive plastic strains (Figs. 6.8a and 6.8b) (31). Depth of draw (outer rocker cell)/Width of draw (outer rocker cell) – RaDraw (maximum ratio) †0 (6.40) The maximum value of RaDraw is defined by the user. A default value of 1.5 was used in the examples of this chapter. 55) There should be no sudden change in depth of draw (Fig. 6.8b) (32). Changes in depth of draw –DpDrawMx (maximum value) † 0 where DpDrawMx is a maximum value. The user can define this value. A default value of 40 mm was used in the examples of this chapter. 56,57) The upper and lower plates of the inner rocker should be able to avoid die lock (Figs. 4.8, 6.7a and 6.7b) (33,34). ADieMi (minimum value)-(90-A8) † 0 ADieMi (minimum value)- BR_1Ang†0 (6.42) (6.43) (6.41) 205 58) The draw angle for the inner rocker shell should be smaller than a maximum value from stamping requirement (Figs. 6.8a and 6.8b) (35). Draw angle of back rocker – AdrawMx (maximum value) † 0 (6.44) 59) Depth of draw for the inner rocker cell should not be too large relative to its width to avoid excessive plastic strains (Figs. 6.8a and 6.8b) (36). Depth of draw (inner rocker cell)/width of draw (inner rocker cell) – RaDraw (maximum ratio) † 0 (6.45) 60-69) Spring back angle of the plates of rocker must be smaller than a maximum value (Fig. 6.4) (73-82). Spring back angle is the maximum allowable angle between two plates to create a permanent deformation. SpBkAngj - (180 – spBkMi (minimum value)) † 0 j=0,…,9 (6.46) The user can define spBkMi, which is a minimum value. A default value of 20 degrees was used to obtain the results in this chapter. 70,71) Depth of pillar plates should not be too large relative to their width to avoid excessive plastic strains (Fig.4.9) (37,38). Outer_pillar_width/Pillar_outer_ - RaDraw (maximum ratio)†0 (6.47) (6.48) 72) The distance between spot welds and the root of a flange should be larger than a minimum value from manufacturing considerations (Fig. 4.13) (66). Weld_root_Mi(minimum value of weld_root_distance – (length_of_flangespot_weld_placement) † 0 (6.49) Inner_pillar_width/Pillar_inner_length – RaDraw (maximum ratio)† 0 206 The user can define the value of Weld_root_Mi. A default value of 6 mm was used to get the results of examples in this chapter. 73) Spot_weld_spacing should be larger than a minimum value from manufacturing considerations (Fig.4.13) (67). Spot_W_Mi (minimum value of spot_weld_spacing-spot_weld_spacing †0 (6.50) The user can define the value of spot_W_Mi. A default value of 35 mm was used to get the results of examples in this chapter. • Styling Constraints 74) The bottom flange of the rocker should not be visible to a person standing at one side of the car (Fig. 6.9) (39). BR_alpha - BR_beta † 0 (6.51) 75) The slope of the outer rocker cell, A3, should be approximately equal to the slope of the door (Fig. 4.8) (48). |-A3 +Ang_Door| - ABDrMx †0 (6.52) The user can define the values of ABDrMx and Ang_Door. Their default values are 15 degree and 20 degrees, respectively. 207 • Mathematical Constraints 76) The summation of inner_rocker_height and height of BR_L6 should be less than the overall height of rocker (Figs. 4.8 and 6.3) (1). Inner_rocker_height+ vertical distance of L6-rocker_height †0 (6.52) 77) The summation of inner_flange_distance and outboard_cell_width should be less than the overall width of rocker to ensure a meaningful rocker section (Fig. 4.8) (2). Inner_flange_distance + outboard_cell_width - rocker_width †0 (6.53) 78) The summation of low_door_ht_minus_clearance and door_edge_height should be less than overall height of rocker to ensure that a design has a meaningful rocker section (Fig. 4.8) (3). Low_door_ht_minus_clearance + door_edge_height - rocker_height †0 (6.54) 79,80) The inner vertical blending radii must no exceed the pillar height (Fig.4.10) (67). aft_inner_ver_blending_rad - pillar_height † 0 fwd_inner_ver_blending_rad - pillar_height † 0 (6.55) (6.56) 81) The summation of pillar_location, pillar_base, fwd_inner_hor_blending_rad, pillar_inner_length, pillar_outer_length, and rocker_length should be less than the overall length of rocker (Fig. 4.10)(11). Pillar_location+Pillar_base+fwd_inner_hor_blendign_rad+pillar_inner_ length-pillar_outer_length - rocker_length +20 † 0 (6.57) 208 where 20 is small margin to avoid finding a design that can not be created by the parametric CAD model. 82) The afterward inner horizontal blending radius must be smaller than the length of pillar_location (Fig. 4.10) (9). Aft_inner_hor_blending_rad – pillar_location † 0 • Performance Target Constraints 83-85) Typically, the stiffness of a joint must meet two types of requirements: it should not be less than a minimum value and/or should be close to a target value. The first requirement is expresses as follows: ˆ (K I / O − K I / O ) / K I / O ≤ 0 ˆ (K F / A − K F / A ) / K F / A ≤ 0 ˆ (K − K ) / K ≤ 0 Tor Tor Tor (6.58) (6.59) (6.60) (6.61) 6.3 Results and Discussion Above optimization problem was solved several times by changing the design requirements for a joint using nonlinear optimization program DOT (1995). First, optimization results were validated by comparing the stiffness and mass of the obtained joint designs with FEA results. Then, results obtained using response surface polynomial and neural network translators were presented and compared. 6.3.1 Checking the Convergence of the Optimization Program The optimization of B-pillar to rocker joint is a nonlinear problem. The optimizer may converge to a local minimum instead of a global minimum. We can use three methods to check whether the optimization program converges to a global minimum. 209 • • • We solve the problem from different initial points to see if the optimization program converges to the same optimum design. We use the final optimum design as an initial point, and solve the problem again to see if the optimum design changes. We also solve the same problem using different methods to see if the optimization program converges to the same point. In this study, we used the method of modified feasible direction (MFD) and sequential linear programming (SLP). Eight randomly generated designs were used as the initial points. The measured stiffness of an actual car joint was used as the target stiffness. We only considered mass in the objective function, e.g., α in Eq.6.1 was set to be one. Table 6.2 shows the objective functions of the optimum designs when we start from different initial points. We used sequential linear programming (SLP) to obtain the above results. Both RSP and NN translators were tested. According to Table 6.2, the maximum relative difference between the masses of optimum designs is 0.1%. The first 2 to 3 digits of the objective function are the same when starting from different initial designs. For the final design, the first 2 to 3 digits of design variables are found to be the same. Practically no improvement in the objective function (less than 0.1%) was achieved by solving the problem again starting from the obtained optimum designs. We also used modified feasible direction method (MFD) to check the convergence of the optimization program when we start from different designs. We found that the optimum design obtained using MFD was the same as that obtained using SLP method. However, performing optimization using MFD method needed more iterations than using SLP method. SLP was more efficient than MFD for the optimization problems in this study. Because of these reasons, we chose SLP as the default optimization method in the translator B for B-pillar to rocker joint. All the optimization results presented in this chapter were obtained using SLP method. The user can easily change to the MFD method by changing the identification number of the optimization method in the program of translator B. 210 6.3.2 Comparison of Results of Translator B with FEA Results As mentioned earlier, translator B finds the joint with the smallest mass whose stiffnesses are equal to or higher than given target values. To validate translator B, we compared the mass and stiffnesses of optimum designs calculated from translator B and from FEA. To generate some random performance targets (stiffness requirements), we first needed to select ranges in which I/O, F/A, and torsion stiffnesses could vary. We observed that the stiffnesses in different directions were correlated. The user should not ask for a joint that has very high I/O stiffness, but very low F/A, and torsion stiffness because the user may get an optimum design whose stiffness correlates poorly with FEA results. The following is the main reason. Since both the RSP and NN were created using a database obtained using FEA, each stiffness in this database has a range in which it can vary. Translator A is accurate for designs that are close to the center of the feasible domain of the design variables in the database. The predictions from translator A deteriorate when a design approaches the boundary of the database. If the stiffness requirement is high in one direction, but low in another, the optimization program will find designs that are close to the boundary of the region or even outside this region. As a result, translators A and B will not predict the stiffness and mass of the optimum design accurately. To avoid the above problem, we divided the database into several zones by observing the scatter plots of I/O stiffness versus F/A stiffness, and I/O stiffness versus torsion stiffness of the database. Figures 6. 10a and 6.10b show that the I/O, F/A, and torsion stiffness were actually correlated. For example, a design with a large I/O stiffness often has large F/A and torsion stiffness. Therefore, to ensure that optimization results from the translator B correlate well with FEA results, the combination of stiffness targets should be within these zones. Table 6.3 shows eight zones of the database. We have implemented these zones in the translator B for B-pillar to rocker joint. The translator B will warn the user that the optimization results may be invalid if the combination of stiffness targets is outside the zones. We selected a number of random performance targets (stiffness requirements) within the zones. Using translator B, the optimum design corresponding to each set of stiffness 211 requirements was found. Then Pro/Engineer was used to create the models corresponding to the optimum designs, and the NASTRAN bulk data files. We analyzed the model using MSC/NASTRAN to find the stiffnesses and masses of the optimum designs. Table 6.4a and Figures 6.11a-6.11d compare the predictions of translator B for the I/O, F/A, torsion stiffnesses and mass with FEA results when response surface polynomial translators were used. Table 6.4b and Figs. 6.12a-6.12d show the corresponding results for neural network translators. The correlation coefficient is used to measure the correlation between the FEA and the optimum results. Correlation coefficient is: ρY = cov( Y O ,Y FEA ) σ Y ×σ Y O FEA o ,YFEA (6.62) where cov( YO , YFEA ) = 2 σY = O 1 nc ∑ ( yO , j − yO )( y FEA, j − y FEA ) (6.63a) (6.63b) (6.63c) 1 nc = ∑(y O, j − yO ) 2 − y FEA ) 2 2 σY FEA 1 nc ∑( y FEA , j where yO , j and y FEA , j are the optimization and FEA results, respectively. yO and y FEA are the mean values of the optimization and FEA results. nC is the number of designs used in comparison. YO and YFEA are the vectors of the optimization and FEA results, respectively. In general, a correlation coefficient larger than 0.9 shows good agreement between predictions of translator B and FEA results. Note that the square of the correlation coefficient is equal to the coefficient of determination, R2, which was defined in Eq. 2.5 The correlation coefficients for stiffnesses in different directions are given in the Figs. 6.11a - 6.12d and Table 6.4c. The correlation coefficients of RSP translators 212 range from 0.9677 to 0.9920. For the NN translators, the correlation coefficients range from 0.8809 to 0.9810. 6.3.3 Redesign of the Joints of two Cars Using Translator B We also used the translator B to redesign two actual car joints. We set the stiffness targets equal to the stiffnesses of the actual joints. We obtained the optimum designs using the translator B. Tables 6.5a to 6.5d show the optimization and FEA results. From the comparison of the optimization and FEA results of the two joints, we observed the following trends when optimizing the shapes of the joint components: B-pillar • Pillar_outer_length reaches its upper bound at the optimum design. Outer_pillar_width increases when the stiffness requirements increase. Both help increase the cross section of B-pillar and the stiffness. • Pillar_base increases as the stiffness requirements increase. This is reasonable because the stiffness of a joint is expect to increase rapidly as the section modulus of the beam leading to the joint increases. Pillar reinforcement • Because pillar_reinf_base_width and pillar_reinf_expansion only have little effects on the stiffness of the joint, they take values close to their lower bounds to reduce the mass of the joint. Cross section of rocker • Rocker_height and low_door_ht_minus_clearance reach their lower bounds. outboard_cell_width is close to its lower bound. Rocker_width, inner_rocker_height and inner_flange_distance assume values between their lower and upper bounds. • Door_edge_height tends to be as large as possible, and door_edge_width tends to be small as possible. This trend of the translator to minimize the door edge width also has been observed by Zhu (1994). The reason for reducing the 213 door_edge_width is that the stiffness increases rapidly as the door_edge_width decreases. • A1, A3, and A8 reach their lower bounds. A2 and A7 are close to their upper bounds. As a result of these trends, the front part of rocker tends to become vertical, and the bottom of rocker tends to become horizontal. Pillar hole • Among the four dimensions that define the opening of B-pillar, aft_pillar_hole is fixed. Bottom_pillar_hole and fwd_pillar_hole reach their upper bounds. This makes the opening of B-pillar as small as possible. Plate thickness • The value of thickness of front rocker assumes the lower bound for car B and a value between its lower and upper bounds for car A. The parametric study in Section 6.3.4 shows that increasing the thickness of front rocker greatly increases both the mass and stiffness of a design. Therefore, determining a suitable value between the lower and upper bounds is a trade-off for an optimum design between high stiffness and low mass. For smaller stiffness requirements, the thickness of front rocker reaches its lower bounds to reduce the mass. For high stiffness requirements, it takes a value between its lower and upper bounds. • Thicknesses of pillar reinforcement, center plate and back rocker do not have big effects on the stiffness. They reach their lower bounds to reduce the mass of the joint. • Thickness of pillar back increases as the stiffness requirements increase. The study in Section 6.3.4 also shows that increasing the thickness of pillar back increases the stiffness of the joint. Table 6.5d compares the mass of the two joints before and after optimization. It is found that improvements range from 8.1% to 27.5%. It should be noted that we used the bounds measured from actual cars to obtain the above results. In practice, the ranges of the design variables may be smaller, and some design variables may be fixed because of styling, packaging, or manufacturing considerations. This will reduce the region in 214 which the optimizer can search for an optimum design. Therefore, actual improvement of two joints may be smaller than that in Table 6.5d. Figures 6.13 compares the shapes of the initial and the optimum designs. We also observed the 12 designs compared in Section 6.3.2, and found the trends were the same as the above. 6.3.4 Parametric Study Figures 6.14 to 6.16 show the effect of I/O, F/A, and torsion stiffness requirements on the mass of the optimum design. To get the above results, we only changed one stiffness requirement at a time and fixed the other two stiffness requirements at the values corresponding to an actual joint. It is observed that for small stiffness requirements, the mass of the final design is almost constant or increases very little. After a certain point, the mass begins to increase at a considerable rate with the stiffness. It is found that the F/A stiffness requirement has larger effects on the mass than the I/O and torsion stiffness requirements. Thus, for the range of stiffness considered in this project, the mass of the optimum design is mainly determined by the F/A stiffness requirement. The I/O stiffness requirement has larger effects on the mass compared with the torsion stiffness requirement. It is found that some design variables in an optimum design tended to reach their lower or upper bounds, and some others had values between the lower and upper bounds. The mass of the optimum design changes when we change the lower and upper bounds of these design variables. Figures 6.17 to 6.25 show the effects of the lower and upper bounds of some design variables on the mass of the optimum design. The measured stiffness of an actual joint was used as the stiffness targets in this study. The design variables are thickness of front rocker, thickness of pillar back, pillar_base, outer_pillar_width, pillar_inner_length, door_edge_width, rocker_width, and outboard_cell_width. From these figures, we find that the mass of optimum design increases when we increase the lower bound or decrease the upper bound of these design variables because 215 these changes reduce the region in which the optimizer can search for an optimum design. Thickness affects significantly the mass of the optimum design. Figures 6.17-6.18 show the relation between the lower bounds of thickness of the front rocker and pillar back on the mass. The thickness of pillar back of the optimum design contributes comparatively less to the mass while increasing its value significantly increases the stiffness. When we reduce its upper bound, the optimum design tends to increase the thickness of the front rocker to increase the overall stiffness of the joint. Pillar_base is one of the most important design variables. For the optimum design, the value of pillar_base falls between its lower and upper bounds. Figures 6.19 and 6.20 show that the effects of the lower and upper bounds of pillar_base on the mass can be significant. Figures 6.21-6.25 show the effects of the bounds of outer_pillar_width, pillar_inner_length, door_edge_width, rocker_width, and outboard_cell_width on the mass of the optimum design. 6.3.3 Discussion of Results For the B-pillar to rocker joint, we developed the translator B using both the response surface polynomials and neural networks. Both MFD and SLP methods were tested. SLP was chosen as the default optimization algorithm because of its efficiency when applied to the problems considered in this project. The user can easily switch to the MFD method by changing a variable in the program for translator B. From the comparison of the optimum and FEA results, we found that it is important to choose a combination of stiffness targets for which translator B is valid. For this purpose, we defined eight zones in which the stiffness requirements should fall. The results of both NN and RSP translators correlated well with FEA results, except when the stiffness targets were very high (Figs. 6.11a to 6.12d). By comparing the 216 results obtained using RSP and NN, we found that the mass of the optimum design obtained using NN translators was typically about 10% higher than that obtained using RSP translators. This is because the NN tended to underestimate stiffness (Figs 6.12a6.12c). The correlation coefficients between the FEA results and the optimization results obtained using RSP and NN translators were compared in Table 6.4c. We found that the RSP translator B predicted F/A stiffness and mass more accurately than the NN translator B. Predictions for I/O and torsion stiffness from both types of translators were equally accurate. Note that when we used the NN translator, we considered several design variables that do not affect stiffness and mass. More accurate results could be obtained if these variables were fixed. Even if the testing results from RSP translators seem to be slightly better than those from NN translators for the 12 design compared, it is difficult to conclude whether the RSP or the NN translator is better just from the 12 designs we compared. This is different from the Zhu’s conclusion (1994), who found that the predictions of NN were slightly more accurate than those of RSP for a set of 29 randomly generated designs. This is because he used a completed second degree polynomial to create the RSP translator that should include some statistically unimportant terms. As a result, Zhu’s RSP translator performed more poorly than a translator developed using stepwise regression. Some observations about the effects of the important design variables should help designers to improve the joint stiffness, and design more efficient joints. For example, this study found that the value of pillar_base of the optimum design increases as the stiffness requirements increase. This shows that the optimum design tends to make the joint connection between the rocker and the B-pillar as large as possible, which agrees with rule of thumb presented in internal reports of an automotive manufacturer. In most optimum designs, the bottom_pillar_hole and the fwd_pillar_hole tended to reach their upper bounds. This made the pillar hole of the optimum design as small as possible. This is also consistent with rules of thumb presented in an internal report. It was found that pillar_reinf_base_width and pillar_reinf_expansion did not have big effects on the stiffness of the joint. They assumed values equal to their lower bounds. 217 We found that rocker_height and outboard_cell_width reached their lower bounds. This is different from Zhu’s conclusion (1994). In his study, he found that rocker section tended to take the maximum allowable height for the optimum design. This may be caused by the difference of parametric model and constraints between this study and that of Zhu’s. Rocker_width and inner_flange_width took values between their lower and upper bounds. A1, A3, and A8 reached their lower bounds, and A2 and A7 were close to their upper bounds. The front part of rocker tended to become vertical, and the bottom part of rocker tended to become horizontal. This trend increased the area of the rocker cross section and the stiffness of the joint. This study found that when the I/O and F/A stiffness requirements were low, the RSP translator B tended to overestimate the stiffness of the optimum design compared to FEA results. When the stiffness requirements were high, the translator B tended to underestimate stiffnesses (Figs. 6.11a-6.11c). The RSP translator consistently underestimated the mass of the optimum designs (Fig. 6.11d). On the other hand, in general, the NN translator B underestimated stiffnesses. For the 12 designs studied, the NN translator B tended to underestimate F/A, and torsion stiffness compared to the RSP translator B. The NN translator also underestimated the mass of a few designs, but its bias was smaller than that of the RSP translator. Zhu (1994) suggested increasing the minimum allowable stiffness by 10% to ensure translator B yields designs whose stiffness is equal to or exceeds the given target values. In this study, we did not notice the phenomenon that translator B consistently overestimates the design. Therefore, we do not suggest to slightly increase the stiffness targets for the optimum designs. 218 Table 6.1: Ranges and States of Design Variables Seq. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Name of Design Variables Length_of_flange Spot_weld_spacing Spot_weld_placement Pillar_base Pillar_angle Pillar_io_angle Pillar_height Pillar_location Outer_pillar_width Inner_pillar_width Pillar_outer_length Pillar_inner_length Fwd_inner_ver_blending_rad Fwd_inner_hor_blending_rad Fwd_outer_ver_blending_rad Fwd_inner_hor_blending_rad Aft_inner_ver_blending_rad Aft_inner_hor_blending_rad Aft_outer_ver_blending_rad Aft_outer_hor_blending_rad Inner_pillar_base_width Rocker_length Pillar_reinforcement_depth Pillar_reinf_base_width Pillar_reinf_expansion A1 A2 A3 A5 A7 A8 Rocker_height Inner_flange_distance Inner_rocker_height Rocker_width Outboar_cell_width Door_edge_height Door_edge_width Bounds (mm, degree) Lower Upper 19 47.5 9.5 157 90 90 210 150 50 6 77 122 120 80 120 80 95 100 95 100 2 485 10 20 2 83 65 10 75 70 80 110 18 90 115 65 6 7 19 47.5 9.5 215 90 90 250 150 83 25 110 160 155 125 155 125 133 135 140 135 15 485 100 60 15 90 78 36 87 86 90 120 45 110 147 95 38 19 States # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user # Fixed by user ## Dependent # Fixed by user Independent Independent Independent Independent Independent Independent ## Dependent ## Dependent Independent Independent ## Dependent ## Dependent Independent ## Fixed by program ## Dependent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent 219 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Low_door_ht_minus_clearance Fwd_bulk_head_position Aft_bulk_head_position Top_pillar_hole Bottom_pillar_hole Fwd_pillar_hole Aft_pillar_hole Thickness of frontrock Thickness of pillar_reinf Thickness of pillarback Thickness of backrock Thickness of centerplate Thickness of pillar_bridge Thickness of bulkheads 60 470 10 22 28 17 15 0.71 0.71 0.89 1.27 0.71 1.0 1.4 75 470 10 80 50 30 15 1.27 1.52 1.27 1.98 1.27 1.0 1.78 Independent Not used Not used Independent Independent Independent ## Fixed by user Independent Independent Independent Independent Independent Not used Not used ## & #, excluded from optimization Length of B-pillar branch (Fig. 6.1) is fixed at 269 mm. Table 6.2: Comparison of Optimum Results When Starting From Different Initial Points ( K I / O > 4.3899 E 7 Nmm , K F / A > 5.2297 E8 Nmm , K Tor > 7.9788 E 7 Nmm ) Seq. No 1 2 3 4 5 6 7 8 Use Polynomial Translators No. of Function Object Function Evaluations (kg) 527 4.54870 527 4.54962 527 4.54821 492 4.54730 492 4.54704 492 4.54812 527 4.55183 528 4.55183 Use Neural Network Translators No. of Function Objective Function Evaluations (kg) 740 4.97923 740 4.97973 705 4.98170 635 4.98150 600 4.97854 706 4.97915 705 4.98036 673 4.97942 220 Table 6.3: Zones of Database (According to the Combination of Stiffness ) Zone No. 1 2 3 4 5 6 7 8 Range of K I / O ( × 10 7 Nmm) 1.0~1.5 1.5~2.0 2.0~3.0 3.0~4.0 4.0~5.0 5.0~6.0 6.0~7.0 7.0~8.0 Range of K F / A ( × 10 8 Nmm)) 1.8~4.0 2.0~4.5 2.0~5.5 2.0~5.5 2.5~5.5 2.5~5.0 3.5~5.0 3.5~5.0 Range of K Tor ( × 10 7 Nmm) 2.5~8.0 3.0~10.0 3.5~12.5 4.0~13.5 4.0~14.5 5.0~13.0 7.5~13.0 9.0~13.0 Table 6.4a: Comparison of Optimization Results and FEA Results for B-pillar to Rocker Joint (using RSP Translators) Design No Stiffness Requirement (Nmm) F/A Tor I/O ×108 ×107 ×107 I/O ×107 Optimization Results (Nmm, kg) F/A Torsion ×108 ×107 Mass I/O ×107 FEA Results (Nmm, kg) F/A Torsion ×108 ×107 Mass 1 2 3 4 5 6 7 8 9 10 11 12 1.25 1.75 2.5 3.3 3.6 4.3 4.6 5.3 5.6 6.3 6.6 7.5 2.5 3.0 3.5 3.5 4.5 4.0 5.0 3.5 4.0 4.0 4.5 4.5 5.0 6.5 8.0 8.5 10. 9.0 10. 9.0 10. 10. 11. 11. 1.4175 1.7504 2.5003 3.3002 3.6000 4.3000 4.6000 5.3005 5.6002 6.3000 6.6000 7.4996 2.5000 3.0001 3.5002 3.5000 4.5000 4.0000 5.0000 3.4999 4.0000 4.0000 4.5000 4.5000 4.9996 6.5001 8.0007 8.5000 10.001 9.0000 12.168 9.0004 10.001 10.000 14.260 15.379 3.4099 3.5445 3.6865 3.7138 3.9397 3.8364 4.3960 3.8585 3.9898 4.0835 4.3461 4.4949 1.0990 1.5428 2.0414 2.5715 3.1401 4.4447 5.1175 5.1304 5.2545 7.2950 8.5703 9.9149 2.3485 2.8517 3.0754 3.2920 4.3361 4.2336 5.4079 3.4553 3.7779 4.3145 4.9361 4.9618 4.1962 6.0755 9.0905 9.6463 11.245 10.483 14.206 9.2918 11.264 10.337 15.774 16.824 3.6599 3.7395 3.8463 3.8601 4.0734 3.9484 4.5069 3.9485 4.0638 4.2581 4.4874 4.6146 221 Table 6.4b: Comparison of Optimization Results and FEA Results for B-pillar to Rocker Joint (using NN Translators) Design No Stiffness Requirement (Nmm) F/A Tor I/O ×107 ×107 ×107 I/O ×107 Optimization Results (Nmm, kg) F/A Torsion ×108 ×107 Mass I/O ×107 FEA Results (Nmm, kg) F/A Torsion ×108 ×107 Mass 1 2 3 4 5 6 7 8 9 10 11 12 1.25 1.75 2.5 3.3 3.6 4.3 4.6 5.3 5.6 6.3 6.6 7.5 2.5 3.0 3.5 3.5 4.5 4.0 5.0 3.5 4.0 4.0 4.5 4.5 5.0 6.5 8.0 8.5 10. 9.0 10. 9.0 10. 10. 11. 11. 2.9688 3.7676 3.8915 4.0418 3.6807 4.3000 4.6001 5.3004 5.6001 6.3001 6.6000 7.5000 3.2061 3.3338 3.5000 3.5000 4.4998 4.0000 5.0001 3.5000 4.0000 4.0000 4.5000 4.4999 5.6278 6.5004 7.9997 8.4997 11.995 11.392 13.186 8.9996 12.116 12.644 14.216 14.561 3.9192 3.9235 3.9540 3.9676 4.4386 4.1179 4.8121 3.9917 4.1656 4.2252 4.5564 4.7961 3.2403 5.1058 5.1183 5.1352 3.4885 5.0931 4.8263 6.9335 6.7701 7.8590 9.0910 9.6724 3.5276 4.4350 4.4917 4.5034 5.0269 5.1805 5.7996 4.1704 4.6281 4.5805 5.2361 4.8660 6.9522 8.3798 9.8755 10.284 13.593 12.860 15.905 11.130 14.047 15.256 18.610 18.543 3.8604 3.9105 3.9556 3.9627 4.4863 4.1348 4.8888 4.1492 4.2264 4.3893 4.7582 4.8464 Table 6.4c: Comparison of Correlation Coefficients Obtained using RSP and NN translators Stiffness/Mass I/O F/A Torsion Mass RSP translators 0.9733 0.9677 0.9872 0.9920 NN translator 0.9703 0.8809 0.9831 0.9810 222 Table 6.5a: States of Design Variables and Optimum Designs for Car A No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Name of Design Variables Length_of_flange Spot_weld_spacing Spot_weld_placement Pillar_base Pillar_angle Pillar_io_angle Pillar_height Pillar_location Outer_pillar_width Inner_pillar_width Pillar_outer_length Pillar_inner_length Fwd_inner_ver_blending_rad Fwd_inner_hor_blending_rad Fwd_outer_ver_blending_rad Fwd_inner_hor_blending_rad Aft_inner_ver_blending_rad Aft_inner_hor_blending_rad Aft_outer_ver_blending_rad Aft_outer_hor_blending_rad Inner_pillar_base_width Rocker_length State # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user # Fixed by user ## Dependent # Fixed by user Independent Independent Independent Independent Independent Independent ## Dependent ## Dependent Independent Independent ## Dependent ## Dependent Independent ## Fixed by program Pillar_reinforcement_depth ## Dependent Pillar_reinf_base_width Independent Pillar_reinf_expansion Independent A1 Independent A2 Independent A3 Independent A5 Independent A7 Independent A8 Independent Rocker_height Independent Inner_flange_distance Independent Inner_rocker_height Independent Rocker_width Independent Outboar_cell_width Independent Door_edge_height Independent Door_edge_width Independent Low_door_ht_minus_clearanc Independent e 223 Car A (mm, degree) RSP NN 19.00 19.00 47.50 47.50 9.50 9.50 185.15 185.00 90.00 90.00 90.00 90.00 218.45 218.37 150.00 150.00 82.99 83.00 6.00 23.00 109.99 110.00 159.84 159.99 120.00 134.99 80.00 80.00 120.00 134.99 80.00 80.00 132.99 133.00 135.00 100.00 132.99 133.00 135.00 100.00 15.00 2.32 485.00 485.00 38.16 20.00 2.00 83.00 77.94 10.00 75.00 85.95 80.03 110.01 37.57 106.16 120.17 67.83 35.66 7.00 60.05 46.82 20.00 2.00 83.00 77.99 10.00 75.00 85.95 80.01 110.01 36.94 106.19 119.75 68.03 27.07 7.00 60.00 40 41 42 43 44 45 46 47 48 49 50 51 52 Fwd_bulk_head_position Aft_bulk_head_position Top_pillar_hole Bottom_pillar_hole Fwd_pillar_hole Aft_pillar_hole Thickness of frontrock Thickness of pillar_reinf Thickness of pillarback Thickness of backrock Thickness of centerplate Thickness of pillar_bridge Thickness of bulkheads Not used Not used Independent Independent Independent ## Fixed by user Independent Independent Independent Independent Independent Not used Not used 80.00 49.99 30.00 15.00 1.14 0.71 1.27 1.27 0.71 80.00 50.00 30.00 15.00 1.27 0.71 1.27 1.39 0.71 224 Table 6.5b: States of Design variables and Optimum Designs for Car B No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Name of Design Variables Length_of_flange Spot_weld_spacing Spot_weld_placement Pillar_base Pillar_angle Pillar_io_angle Pillar_height Pillar_location Outer_pillar_width Inner_pillar_width Pillar_outer_length Pillar_inner_length Fwd_inner_ver_blending_rad Fwd_inner_hor_blending_rad Fwd_outer_ver_blending_rad Fwd_inner_hor_blending_rad Aft_inner_ver_blending_rad Aft_inner_hor_blending_rad Aft_outer_ver_blending_rad Aft_outer_hor_blending_rad Inner_pillar_base_width Rocker_length Pillar_reinforcement_depth Pillar_reinf_base_width Pillar_reinf_expansion A1 A2 A3 A5 A7 A8 Rocker_height Inner_flange_distance Inner_rocker_height Rocker_width Outboar_cell_width Door_edge_height Door_edge_width Low_door_ht_minus_clearanc e Fwd_bulk_head_position Car B (mm, degree) RSP NN # Fixed by user 19.00 19.00 # Fixed by user 47.50 47.50 # Fixed by user 9.50 9.50 Independent 157.15 160.24 # Fixed by user 90.00 90.00 # Fixed by user 90.00 90.00 ## Dependent 218.72 217.62 # Fixed by user 150.00 150.00 Independent 79.34 50.00 Independent 7.54 24.99 Independent 109.93 109.96 Independent 122.07 159.89 Independent 120.01 155.00 Independent 125.00 80.03 ## Dependent 120.01 155.00 ## Dependent 125.00 80.03 Independent 113.06 133.00 Independent 135.00 100.00 ## Dependent 113.06 133.00 ## Dependent 135.00 100.00 Independent 14.99 14.99 ## Fixed by program 485.00 485.00 ## Dependent 43.17 47.88 Independent 20.00 25.00 Independent 2.00 10.00 Independent 83.00 83.00 Independent 77.94 78.00 Independent 10.00 10.00 Independent 75.00 80.12 Independent 81.21 85.98 Independent 80.04 80.00 Independent 110.05 110.01 Independent 40.28 29.93 Independent 102.59 105.81 Independent 120.10 119.64 Independent 65.05 74.93 Independent 30.73 26.65 Independent 7.00 7.00 Independent 60.05 60.01 Not used 225 470.00 470.00 State 41 42 43 44 45 46 47 48 49 50 51 52 Aft_bulk_head_position Top_pillar_hole Bottom_pillar_hole Fwd_pillar_hole Aft_pillar_hole Thickness of frontrock Thickness of pillar_reinf Thickness of pillarback Thickness of backrock Thickness of centerplate Thickness of pillar_bridge Thickness of bulkheads Not used Independent Independent Independent ## Fixed by user Independent Independent Independent Independent Independent Not used Not used 10.00 22.00 49.96 30.00 15.00 0.71 0.71 0.89 1.27 0.71 1.00 1.78 10.00 79.94 47.60 29.97 15.00 0.71 0.71 0.89 1.27 0.71 1.00 1.78 Table 6.5c: Comparison of FEA Results and Results from Translator B for B-pillar to Rocker Joint Cars/ Stiff. Requirements (Nmm) Car A I/O>4.76E7 F/A >5.684E8 Tor >7.649E7 Car B I/O>2.567E7 F/A>2.527E8 Tor > 6.521E7) Stiffness / Mass I/O F/A Torsion Mass I/O F/A Torsion Mass RSP Translators (Nmm, kg) FEA Translator (err) 5.4527E7 4.7600E7 (-12.7%) 6.2650E8 5.6840E8 (-9.3%) 1.6285E8 1.3799E8 (-15.3%) 5.0450 4.9475 (-1.9%) 1.9929E7 2.5674E7 (28.8%) 2.4668E8 2.5270E8 (2.4%) 5.7217E7 6.5211E7 (14.0%) 3.7179 3.5204 (-5.3%) FEA 4.5713E7 6.6915E8 1.7093E8 5.4505 4.0275E7 3.6429E8 8.6064E7 3.9322 NN Translators (Nmm, kg) Translator (err) 4.7602E7 (4.1%) 5.6840E8 (-15.1%) 1.3924E8 (-18.5%) 5.3570 (-1.7%) 3.6564E7 (-9.2%) 3.2990E8 (-9.4%) 6.5210E7 (-24.2%) 3.9231 (-0.2%) Table 6.5d: Comparison of Mass of the Initial and Optimum Designs for Car A and Car B Cars Initial Design (kg) 5.9312 5.1268 Optimum Design (kg) and Percentage Improvement RSP NN 5.0450 (14.9%) 5.4505 (8.1%) 3.7179 (27.5%) 3.9322 (23.3%) Car A Car B 226 Length of B-pillar branch Center of rocker center line Figure 6.1: Length of B-pillar Branch 227 Pillar reinforcement intersects the front part of rocker Pillar reinforcement intersects the bottom part of rocker Figure 6.2: Constraints on the Pillar Reinforcement 228 BR_L8 BR_L1 BR_L2 BR_L7 BR_L3 BR_L4 BR_L6 BR_L5 Figure 6.3: Manufacturing Constraints on the Length of Rocker Cross Section 229 BR_1Ang BR_Angle1 BR_Angle8 BR_Angle2 BR_Angle3 A6 BR_Angle4 BR_Angle7 BR_Angle6 BR_Angle5 Figure 6.4: Manufacturing Constraints on Angles of Rocker Cross Section 230 h4 d7 Ang_Door A3 Figure 6.5: Constraints on the Rear Plate of Rocker and Angle of Outer Rocker Shell 231 spBkAng9 spBkAng0 spBkAng8 spBkAng1 spBkAng2 spBkAng3 spBkAng7 spBkAng4 Maximum angle spBkAng6 spBkAng5 Figure 6.6: Spring Back Angles on Rocker Cross Section 232 Moving direction of die Angles to avoid die lock Die lock angle: Angle1-90 Angle2 -90 Average die lock angle: (Angle1+Angle2-180)/2 Die Sheet metal Base Angle 1 Angle 2 Figure 6.7a: Explanation of Die Lock Angle 233 90-A1 A1 BR_1Ang 90-A8 90-A2 A8 A2 Figure 6.7b: Die Lock Angles on the Rocker Cross Section 234 Draw angle = (Angle 2- Angle 1 +180)/2-90 =(Angle2-Angle1)/2 D’ E’ A’ B’ A C’ O B AA // A’O BB // B’O AB // C’O AB A D’O ³A’OE’ = ³E’OB’ Angle 1 Angle 2 A B Figure 6.8a: Explanation of Draw Angle 235 Change in depth of draw D D D’ D’ C’ Draw angle of back rocker C’ C A A Draw angle of front rocker Width of draw (inner/outer rocker cell) A’ A’ B’ B’ DD // D’D’ CC // C’C’ C F O F’ FO = OF’ Depth of draw (outer rocker cell) B B AA // A’A’ BB // B’B’ Depth of draw (inner rocker cell) Figure 6.8b: Constraints on Draw Angle and Draw Steps 236 Observer’s eye BR_alpha BR_beta Figure 6.9: Constraint on the Cross Section of Rocker 237 5 .0 E + 0 8 4 .5 E + 0 8 4 .0 E + 0 8 F /A S tiffn e ss 3 .5 E + 0 8 3 .0 E + 0 8 2 .5 E + 0 8 2 .0 E + 0 8 1 .5 E + 0 8 1 .0 E + 0 7 2 .0 E + 0 7 3 .0 E + 0 7 4 .0 E + 0 7 5 .0 E + 0 7 6 .0 E + 0 7 7 .0 E + 0 7 8 .0 E + 0 7 I/ O S ti ffne s s Figure 6.10a: Correlation Between I/O and F/A Stiffness for Designs in the Database 238 1 .4 E + 0 8 1 .2 E + 0 8 1 .0 E + 0 8 T o r s io n S t i f f n e s s 8 .0 E + 0 7 6 .0 E + 0 7 4 .0 E + 0 7 2 .0 E + 0 7 1 .0 E + 0 7 2 .0 E + 0 7 3 .0 E + 0 7 4 .0 E + 0 7 5 .0 E + 0 7 6 .0 E + 0 7 7 .0 E + 0 7 8 .0 E + 0 7 I/ O S ti ffne s s Figure 6.10b: Correlation Between I/O and Torsion Stiffness for Designs in the Database 239 1.0E+08 8.0E+07 11 10 12 Optimization Results 6.0E+07 8 6 9 7 4.0E+07 4 3 5 Use polynomial translators Correlation coefficient =0.9733 2.0E+07 2 1 0.0E+00 0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08 FEA Results Figure 6.11a: Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the I/O Stiffness of B-pillar to Rocker Joint 240 6.0E+08 7 5.0E+08 Optimization Results 5 11 12 4.0E+08 3 4 8 9 10 6 Use polynomial translators 2 3.0E+08 1 Correlation coefficient =0.9677 2.0E+08 2.0E+08 3.0E+08 4.0E+08 FEA Results 5.0E+08 6.0E+08 Figure 6.11b: Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the F/A Stiffness of B-pillar to Rocker Joint 241 2.0E+08 1.6E+08 11 12 Optimization Results 7 1.2E+08 10 8 4 3 2 1 6 5 9 Use polynomial translators Correlation coefficient =0.9872 8.0E+07 4.0E+07 4.0E+07 8.0E+07 1.2E+08 FEA Results 1.6E+08 2.0E+08 Figure 6.11c: Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the Torsion Stiffness of B-pillar to Rocker Joint 242 5 4.5 11 12 7 Optimization Results 10 4 8 6 4 3 2 9 5 Use polynomial translators Correlation coefficient =0.9920 3.5 1 3 3 3.5 4 FEA Results 4.5 5 Figure 6.11d: Comparison of FEA Results and the Optimization Results Obtained using RSP Translators for the Mass of B-pillar to Rocker Joint 243 1 .0 E + 0 8 8 .0 E + 0 7 11 12 O p tim iz a tio n R e su lts 10 6 .0 E + 0 7 7 9 8 6 4 3 2 4 .0 E + 0 7 1 5 Use neural network translators Correlation coefficient =0.9703 2 .0 E + 0 7 0 .0 E + 0 0 0 .0 E + 0 0 2 .0 E + 0 7 4 .0 E + 0 7 6 .0 E + 0 7 8 .0 E + 0 7 1 .0 E + 0 8 F E A R e s ul ts Figure 6.12a: Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the I/O Stiffness of B-pillar to Rocker Joint 244 6 .0 E + 0 8 7 5 .0 E + 0 8 O p tim iz a tio n R e su lts 12 5 11 10 9 6 4 .0 E + 0 8 8 1 3 4 2 3 .0 E + 0 8 Use neural network translators Correlation coefficient =0.8809 2 .0 E + 0 8 2 .0 E + 0 8 3 .0 E + 0 8 4 .0 E + 0 8 F E A R e s ul ts 5 .0 E + 0 8 6 .0 E + 0 8 Figure 6.12b: Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the F/A Stiffness of B-pillar to Rocker Joint 245 2 .0 E + 0 8 1 .6 E + 0 8 12 O p tim iz a tio n R e su lts 11 7 5 9 10 1 .2 E + 0 8 6 8 4 Use neural network translators Correlation coefficient =0.9831 8 .0 E + 0 7 2 1 3 4 .0 E + 0 7 4 .0 E + 0 7 8 .0 E + 0 7 1 .2 E + 0 8 F E A R e s ul ts 1 .6 E + 0 8 2 .0 E + 0 8 Figure 6.12c: Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the Torsion Stiffness of B-pillar to Rocker Joint 246 5 12 7 11 4 .5 O p tim iz a tio n R e s u lts 10 5 9 6 4 3 1 2 4 8 Use neural network translators 3 .5 Correlation coefficient =0.9810 3 3 3 .5 4 F E A R e s u lts 4 .5 5 Figure 6.12d: Comparison of FEA Results and the Optimization Results Obtained using NN Translators for the Mass of B-pillar to Rocker Joint 247 Rocker Section of Initial Design (dashed line) Rocker Section of Optimum Design (solid line) Figure 6.13: Comparison of Rocker Cross Sections of the Initial Design and Optimum Design 248 7 6.5 P o ly n o mia l R e s u lts 6 5.5 M as s N N R e s u lts 5 4.5 4 3.5 3 1.0E+ 07 2.0E+ 07 3.0E+ 07 4.0E+ 07 5.0E+ 07 6.0E+ 07 7.0E+ 07 8.0E+ 07 9.0E+ 07 I/O S tiffn e s s R e q u ire me n t Figure 6.14: Relation Between Mass and I/O Stiffness Requirement for B-pillar to Rocker Joint (KF/A>5.2297E8, Ktor>7.9788E7) 249 7 6.5 6 5.5 M as s P o ly n o mia l R e s u lts N N R e s u lts 5 4.5 4 3.5 3 2.0E+ 08 2.5E+ 08 3.0E+ 08 3.5E+ 08 4.0E+ 08 4.5E+ 08 5.0E+ 08 5.5E+ 08 6.0E+ 08 F /A S tiffn e s s R e q u ire me n t Figure 6.15: Relation Between Mass and F/A Stiffness Requirement for B-pillar to Rocker Joint (KI/O>4.3899E7, Ktor>7.9788E7) 250 7 6.5 Polynomial results NN results 6 5.5 Mass 5 4.5 4 3.5 3 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08 1.2E+08 1.4E+08 1.6E+08 T orsion Stiffness Requirement Figure 6.16: Relation Between Mass and Torsion Stiffness Requirement for B-pillar to Rocker Joint (KI/O>4.3899, KF/A>5.2297E7) 251 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 0 .8 0 .8 5 0 .9 0 .9 5 1 1 .0 5 1 .1 1 .1 5 1 .2 Lo w e r B o u n d o f T h ic kn e s s o f F ro n t R o c ke r Figure 6.17: Relation Between the Lower Bound of Thickness of Front Rocker and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 252 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 0 .9 5 1 1 .0 5 1 .1 1 .1 5 1 .2 Lo w e r B o u n d o f T h ic kn e s s o f P illa r B a c k Figure 6.18: Relation Between the Lower Bound of Thickness of Pillar Back and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 253 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 155 165 175 185 195 205 Lo w e r B o u n d o f P illa r B a s e Figure 6.19: Relation Between the Lower Bound of Pillar_base and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 254 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 160 170 180 190 200 210 U p p e r B o u n d o f P illa r B a s e Figure 6.20: Relation Between the Upper Bound of Pillar_base and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 255 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 55 60 65 70 75 80 U p p e r B o u n d o f O u te r_ p illa r_ w id th Figure 6.21: Relation Between the Upper Bound of Outer_pillar_width and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 256 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 130 135 140 145 150 155 U p p e r B o u n d o f P illa r_ in n e r_ le n g th Figure 6.22: Relation Between the Upper Bound of Pillar_inner_length and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 257 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 8 9 10 11 12 13 14 15 16 17 18 Lo w e r B o u n d o f D o o r_ e d g e _ w id th Figure 6.23: Relation Between the Lower Bound of Door_edge_width and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 258 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 125 127 129 131 133 135 137 139 141 143 145 Lo w e r B o u n d o f R o c ke r_ w id th Figure 6.24: Relation Between the Lower Bound of Rocker_width and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 259 7 6 .5 P o ly n o mia l R e s u lts 6 N N R e s u lts 5 .5 M as s 5 4 .5 4 3 .5 3 70 72 74 76 78 80 82 84 86 88 90 Lo w e r B o u n d o f O u tb o a rd _ c e ll_ w id th Figure 6.25: Relation Between the Lower Bound of Outboard_cell_width and the Mass of Optimum Design of B-pillar to Rocker Joint (KI/O>4.3899E7, KF/A>5.2297E8, KTor>7.9788E7) 260 Chapter 7 Developing Translator B for an Actual A-pillar to Roof Rail Joint 7.1 Introduction This chapter applies the general methodology for developing translator B in Chapter 3 to an A-pillar to roof rail joint. The A-pillar to roof rail joint is a “Y” shape joint, located on the upper left and upper right sides of the driver and the passenger, respectively (Fig.1.1). The A-pillar joint has three branches. The branch that is perpendicular to the side of the car is called header. The branch that is parallel to the top of the front door is called roof rail. A-pillar is the branch that lies between the windshield and the front door. A-pillar joint supports the windshield of a car, the roof and part of the front door. Chapter 5 describes this joint in detail. This chapter is organized as follows: • Section 7.2 formulates the optimization problem used in developing translator B for the A-pillar to roof rail joint. This section explains the selection and classification of design variables, the objective function and the constraints. • Section 7.3 presents some applications of translator B to some actual design problems. First, we use several methods to check the convergence of translator B. Then, we compare the optimization results with the FEA results. We also use translator B to redesign two actual joints. Since both response surface polynomials and neural networks are used in developing translator B, we also compare the results of both translators. Finally, we perform a parametric study to examine the effects of stiffness requirements and the lower and upper bounds of several important design variables on the mass of the optimum design. Design guidelines for improving the design of the A-pillar to roof rail joint are presented and discussed. 261 7.2 Formulation of the Optimization Problem for Developing a Translator B for an A-pillar to Roof Rail Joint 7.2.1 Definition of the Problem Translator B finds the most efficient design that meets given performance targets (stiffness requirements), and satisfies packaging, manufacturing, styling and performance constraints. Translator B uses optimization. The optimization program searches through the feasible domain to find the most efficient design that satisfies the constraints and performance targets. During each iteration of the optimization process, a new design is analyzed to get its stiffness and mass. It would be very time-consuming to use FEA to get these responses. Moreover, the intermediate designs created in the optimization process are not always feasible, and can not be generated using a generic CAD model. To circumvent these problems, translator A, which was developed in Chapter 5, is used to predict the stiffnesses and masses of new designs during the optimization process. Translator A is encoded in the program of translator B. Translator B uses two types of translators A, a response surface polynomial (RSP) translator A and a neural network (NN) translator A. Chapter 5 describes in detail the translator A for the A-pillar to roof rail joint. There are different types of construction for the A-pillar to roof rail joints. As Chapter 5 explained, different construction types have different reinforcements. Some reinforcements are more common than others. For example, an A-pillar reinforcement that follows the shape of the upper plate of the A-pillar (part 7 in Fig. 5.2) is common. After consulting with engineers from an automotive company, we decided to consider only one type of construction shown in Fig. 5.2. This type of A-pillar to roof rail joint consists of the following parts: the roof (part 1), the top plate of the A-pillar joint (part 2), the bottom plate of the A-pillar joint (part 3), the bottom plate of the header (part 4), and the A-pillar reinforcement (part7). The longitudinal plates that connect two flanges at the 262 three sections of the A-pillar joint (part 5), and the header reinforcement (part 6) are not considered. Translator B has 48 design variables. 7.2.2 Design Variables Table 7.1 shows the design variables for the A-pillar to roof rail joint. As explained in Chapter 3, design variables are divided into four types: 1) Design variables that are fixed by the program because of design rules or conventions. 2) Design variables that are fixed by the user. 3) Dependent design variables whose values can be expressed in terms of other design variables. 4) Independent design variables that can change in optimization. As explained in Chapter 5, some design variables, such as the widths of flanges, and the distance between two adjacent spot welds, are almost always fixed because of manufacturing requirements. These variables are fixed because the user is not expected to change these variables. Values of some design variables, such as the orientation of Apillar, are determined from styling requirements. For a particular joint, the lengths of its three branches are generally fixed because of styling and manufacturing considerations. This allows us to compare the stiffnesses of different joint designs fairly. The user fixes the values of these variables in the input file. These variables do not change in optimization. Six design variables are fixed by the program, namely, distance_from_edge, flange1_width, flange2_width, flange3_width, spot_weld_spacing (Fig. 5.7) and ap_flange1_angle (Fig.5.10). The values of the first five design variables are determined from manufacturing considerations. Flange1_width, flange2_width, and flange3_width are fixed at 20 mm, 16 mm, and 16 mm, respectively. Spot_weld_spacing and distance_from_edge are fixed at 50 mm and 8 mm, respectively. As mentioned in Section 5.2.1, Ap_flange1_angle is set to be zero. This ensures that stiffnesses are defined in the same way for different joint designs, which allows us to compare the stiffnesses fairly. 263 There are two design variables of the second type (fixed by the user), namely, theta, and phi. Because the orientation of A-pillar is determined from styling requirements, theta and phi are fixed at 23 degrees and 76 degrees, respectively. These two values were determined by examining hardware joints. Unfortunately, there is no consistent definition of the center of a joint. In practice, the lengths of the joint branches vary because there is no widely acceptable definition of a reference point that should be used to measure the length of a branch. Indeed, we found a considerable variation in the lengths of the A-pillar, header and roof rail among several hardware joints. It is important, to have a consistent set of reference points for measuring the lengths of all the joint branches and to fix these lengths so that we can compare fairly alternative joint designs. Moreover, we need to define the orientations of the planes of the sections at the ends of the three branches. Figure 7.1 shows a proposed definition of the lengths of the branches of the A-pillar to roof joint. First, we define reference point, O, which is at the intersection of two straight lines tangent to the bases of the outboard flanges of the A-pillar and the roof rail. The length of the header is the distance from point O to the end of the header. A reference point is similarly defined for measuring the lengths of the A-pillar and the roof rail. Figure 7.1 shows that this is the intersection of the lines tangent to the roots of the flanges of the header and the A-pillar. We also defined the planes of the sections at the ends of the three branches to be normal to axes of these members. The axis of each member is tangent to one of the flanges. Specifically, the axis of the A-pillar is tangent to the outboard flange of the A-pillar. The axis of the roof rail is tangent to the outboard flange of the roof rail. The axis of the header is tangent to the forward flange of the header (Fig.5.4) According to our measurements on hardware joints, we fixed the lengths of the header, the roof rail and the A-pillar at 260 mm, 213 mm and 161 mm, respectively. We observe from Fig. 7.1 that the A_pillar_offset, Roof_rail_offset and Header_offset are 264 dependent variables because they can be expressed in terms of the lengths of the branches and other independent variables. Table 7.1 classifies the design variables. The optimization program does not directly change these design variables that are fixed by the program or by the user, or are dependent. Only 37 independent design variables are considered in the translator B. The states and ranges of the design variables are shown in Table 7.1. 7.2.3 Objective Function The objective function of the translator B for the A-pillar to roof rail joint can be expressed as ˆ ˆ ˆ F = αM + (1 − α ) (( K I / O − K I / O ) / K I / O ) 2 + (( K F / A − K F / A ) / K F / A ) 2 + (( K Tor − K Tor ) / K Tor ) 2 where α is a weighting factor. M is the mass of the joint. K I / O , K F / A , and K Tor ˆ ˆ ˆ I/O, F/A and torsion stiffnesses of the joint. K , K , and K are the user-specified I /O F/A Tor (7.1) are the requirements (targets) for I/O, F/A and torsion stiffness. K I / O , K F / A , and K Tor are the stiffness values used to normalize the stiffness. Generally, mass varies from 2 to 8 kg. Each stiffness term varies from 0 to 3. We normalize the stiffness so that the mass and the stiffnesses can have compatible magnitude. In this study, α is set to be 1. Only mass is considered in the objective function. However, α can assume any value between 0 and 1. In this case, the objective function will be the sum of mass and some measure of difference between the stiffness of a design and the given stiffness targets. The optimum design obtained using this objective function will be a joint that not only has low mass but also its stiffnesses are close to given targets. 7.2.4 Constraints The constraints are divided into five types, including packaging, manufacturing, styling, mathematical, and performance constraints. 265 Packaging constraints are related to the arrangement of the car components in space. Manufacturing constraints are due to manufacturing limitations. Styling constraints are due to styling requirements. Mathematical constraints are used to control the ranges of different design variables to ensure that a design has a feasible geometry. Performance constraints dictate that the stiffnesses should exceed given minimum values. The constraints used in translator B are explained in the following section. The number in the parenthesis of each constraint is the corresponding constraint number in the program for translator B. • Packaging Constraints 1) H_blending_rad must be larger than the overall width of roof rail so that the header does not intersect roof rail branch (Figs. 5.6, 5.9, and 7.2) (2). Width of RR cross section +flange3_width-h_blending_rad †0 (7.2) 2) RR_blending_rad should be larger than the overall width of header cross section to prevent the two branches from intersecting (Figs.5.5, 5.6, and 5.8)(3). H_width+flange3_width-header_horizontal_offset + flange3_width RR_blending_rad †0 (7.3) 3-5) The cut distances for parts 2 and 7, and the extension distance of part 4 at the header should be smaller than the overall width of roof rail cross section to avoid intersecting the roof rail branch (Figs. 5.6, 5.12, 5.14, 5.15and 7.2) (7-9). Width of RR cross section –(header_offset - Part2_cut_distance) †0 (7.4) 266 Width of RR cross section –(header_offset-Part4_extension_distance) † 0 (7.5) Width of RR cross section – (header_offset - Part7_cut_distance) † 0 (7.6) 6) The cut distance for part 3 should be smaller than the extension distance of part 4 in order for the two parts to be welded at the bottom (Figs. 5.13 and 5.14)(10). Part3_cut_distance - part4_extension_distance †0 7,8) The edge at the bottom of roof rail section, RLV2, must be in a range (Fig.7.2) (19,20). Minimum value of RLV2 - RLV2 † 0 RLV2- maximum value of RLV2 † 0 (7.8) (7.9) (7.7) 10,11) The angle defining the orientation of flange2 of the roof rail section, FA2, should be in the range [0,90] to accommodate door (Fig.7.3) (13,14). -FA2†0 FA2 – 90 † 0 12,13) RR_h1 should vary in a range (Fig. 5.9) (107,108). Minimum value of RR_h1 - RR_h1 † 0 RR_h1- maximum value of RR_h1 †0 14,15) RR_h2 should vary in a range (Fig. 5.9) (109,110). Minimum value of RR_h2 - RR_h2 † 0 RR_h2 - Maximum value of RR_h2 †0 (7.14) (7.15) (7.12) (7.13) (7.10) (7.11) 267 16) The vertical projection of the door thickness at the roof rail section must be nonnegative (Fig. 7.2) (18). - Roof_rail_height1 † 0 (7.16) 17,19) A-pillar reinforcement must be able to fit into the outer shell of the Apillar (Figs. 5.10 and 5.11) (21-23). Flange1_angle_up - AP_reinf_flange1_angle †0 Flange2_angle_up - AP_reinf_flange2_angle † 0 AP_reinf_depth - AP_window_depth † 0 (7.17) (7.18) (7.19) 20,21) The small edge at the bottom of the A-pillar section should vary in a range (Fig. 7.4) (27,28). Minimum value of ALV2 - ALV2 † 0 ALV2 - maximum value of ALV2 † 0 22,23) The angle defining the orientation of flange2 at A-pillar section, AP_2angle, should be in the range [0, 90] to accommodate door (Fig.7.5) (29,30). -AP_2angle†0 AP_2angle-90.0†0 (7.22) (7.23) (7.20) (7.21) 24) The vertical projection of the door thickness at the A-pillar section must be nonnegative (Fig. 7.4) (26). -A_pillar_door_allowance † 0 (7.24) 268 25,26) The A-pillar reinforcement should not intersect with outer shell of the Apillar section (Fig. 7.6) (31,32). Case 1: Inner plate of A-pillar reinforcement intersects the top plate of Apillar pillar (7.25) (7.26) Case 2: Outer plate of A-pillar reinforcement intersects the top plate of A- The equations corresponding to these cases are omitted because they are too complicated. 27,28) Header_offset should not be too large or too small (Fig. 5.6) (101,102). Minimum value of Header offset- header_offset †0 Header_offset - maximum value of header_offset †0 29,30) Header_horizontal_offset must be in a range (Fig. 5.5) (103,104). Minimum value of header_horizontal_offset- header_horizontal_offset †0 (7.29) Header_horizontal_offset-maximum value of header_horizontal_offset †0 (7.30) 31,32) Roof_rail_offset should be in a range so that the roof rail branch is not too long or too short (Fig. 5.6) (105,106). Minimum value of roof_rail_offset - roof_rail_offset †0 Roof_rail_offset - maximum value of roof_rail_offset † 0 (7.31) (7.32) (7.27) (7.28) 269 33,34) A_pillar_offset should be within a range so that the A-pillar branch is not too long or too short (Fig. 5.6) (111,112). Minimum value of A_pillar_offset – A_pillar_offset†0 A_pillar_offset - maximum value of A_pillar_offset † 0 35,36) There are minimum and maximum values for angleback (Fig. 5.9) (113,114). Minimum value of angleback – angleback † 0 Angleback - maximum value of angleback †0 (7.35) (7.36) (7.33) (7.34) 37,38) The distance between the windshield and the door should not be too large or too small (Fig. 5.10) (115,116). Minimum value of AP_door_ws_distance - AP_door_ws_distance †0 (7.37) AP_door_ws_distance - maximum value of AP_door_ws_ distance†0 (7.38) 39) The ratio of the vertical distance of the vertex to the line connecting the two flanges of A-pillar cross section over the length of the line should be smaller than a maximum value so that inner plate of the A-pillar fits into a cover (Fig.7.7) (98). APVert/APD2f – APRaMx (maximum value) † 0 (7.39) where APRaMx is a maximum ratio. The user can define its value. A default value of 0.5 was used in this chapter. 270 • Manufacturing Constraints Manufacturing constraints include stamping and welding constraints. One needs very detailed information about a joint to determine if a given design is feasible. This information is not available in the early design stage. This study uses crude equations to check if a design is manufacturable. Stamping constraints include constraints for strains, spring back, and die lock. To ensure that plastic strains are low, we impose low limits on the edges of each section, and the angles between two adjacent plates of a section. We also impose an upper limit on the depth of draw to avoid overstretching a plate (Fig.6.8a). Spring back constraints ensure that a plate can be permanently bent. For this reason, the deformation angle should exceed a minimum value (Fig. 7.8). Constraints on die lock ensure that the die and block can be separated after the plate has been stamped. For this purpose, the vertical walls of the hat-shape section should be slanted away from the center of the section (Fig. 6.7a). 40-44) The length of each edge on the header section should be larger than a minimum value so that the header top and bottom shells can be stamped (Fig. 7.9) (39-43). R_L_Mi (a minimum value) - H_Lj †0 j=1,…,5 (7.40) where R_L_Mi is a minimum length. The user can define this length. A default value of 5 mm was used in the examples of this chapter. 45-47) The angles between two adjacent edges on the header section must be larger than a minimum value to avoid a sharp angle between the plates (Fig. 7.10) (44-46). R_A_Mi (a minimu value)- H_Aj † 0 j=1,2,3 (7.41) where R_A_Mi is a minimum angle. The user can define this angle. A default value of 20 degrees was used in the examples of this chapter. 271 48-53) The length of each edge on the roof rail cross section must be larger than a minimum value because of stamping requirement (Fig. 7.11) (47-52). R_L_Mi (a minimum value) -RR_Lj†0 j =1,…,6 (7.42) 54-60) The angle between two adjacent edges on the roof rail cross section must be larger than a minimum value to avoid sharp angle between the plates (Fig. 7.12) (53-59). R_A_Mi (a minimum value) - RR_Aj † 0 j =1,…,7 (7.43) 61-62) The spring back angles of the roof rail section must be larger than a minimum value (Fig. 7.13) (60-62). This constraint on spring back angle ensures that we can create permanent deformation SpBkMi (a minimum value) - RR_Spj † 0 j=1,2,3 (7.44) The user can define the minimum value SpBkMi. A default value of 10 degrees was used. 63) The angle between the inner and outer sides of part 3 must be large enough to avoid die lock (Fig. 7.13) (63). ADieMi (minimum value) – RR_AD1 † 0 (7.45) where ADieMi is a minimum angle. The user can define this angle. A default value of 3 degrees was used. 64-71) The length of each edge of the A-pillar cross section should be larger than the minimum value because of stamping requirements (Fig. 7.14) (64-71). R_L_Mi (a minimum value)- AP_Li † 0 i =1,…,9 (7.46) 272 72-84) The angle between two adjacent edges on the A-pillar cross section must be larger than a minimum value to avoid bending a plate at a sharp angle (Figs. 7.15 and 7.16). AP_A7 should be smaller than 180 degrees (72-84). R_A_Mi (a minimum value)- AP_Aj †0 AP_A7 † 180 j=1, …,12 (7.47) (7.48) 85-91) The spring back angles of the A-pillar cross section should be larger than a minimum allowable value to create a permanent shape (Fig. 7.17) (85-91). SpBkMi (a minimum value)- AP_Spj † 0 j=1,…, 7 (7.49) 92,93) The angle between the two side edges of the A-pillar should be larger than a minimum value to avoid die lock (Fig. 7.17) (92,93). ADieMi (a minimum value) - AP_ADj † 0 j=2,3 (7.50) 94-96) The distance between the spot welds and the roots of flanges must exceed a minimum value so that the plates of the joint can be welded (Fig. 5.7) (94-96). W_R_DMi (a minimum value) - Flangej_width + Distance_from_edge† 0 j=1,2,3 (7.51) The user can define the minimum value W_R_DMi. A default value of 8 mm was used. 97) Spot_weld_spacing must exceed a minimum value because of manufacturing requirements (Fig. 5.7) (97). Minimum value of spot_weld_spacing- spot_weld_spacing † 0 (7.52) 273 A default value of 35 mm was used for the minimum weld spacing. • Styling Constraints 98,99) The difference between the length of the top piece of the door at the roof rail section and that at the A-pillar section should not be too big because of styling considerations (Fig. 7.18). RAraMi and RAraMx are the minimum and maximum ratios specified by the user. Their default values are 0.5 and 2.0, respectively (99,100). RAraMi*AP_edge - RR_edge † 0 RR_edge - RAraMx *AP_edge†0 100) The A-pillar should be slim and should not obstruct driver’s vision (Fig.7.19) (117). Blind angle – AInvMx (a maximum value) † 0 (7.55) (7.53) (7.54) The user can define the maximum value AInvMx. A default value of 10 degrees was used. • Mathematical Constraints 101) The sum of theta and phi must be larger than 90 degrees to determine the orientation of A-pillar (Figs. 5.5 and 5.6) (1). 90 – (Theta+Phi ) †0 (7.56) 102) H_window_depth must be smaller than the overall height of header (Fig.5.8) (11). H_window_depth - H_height † 0 (7.57) 274 103) The sum of AP_inboard_depth and AP_window_depth must be smaller than the overall height of A-pillar so that the shape of the A-pillar cross section is reasonable (Fig. 5.10) (38). AP_inboard_depth + AP_window_depth- AP_height † 0 (7.58) 104-106) The length of the blending regions of a branch should be smaller than the corresponding offset distance from the origin of the global coordinate system to the cross section of that branch (Figs. 5.5 and 5.6) (4-6). H_blending_rad - Header_offset†0 RR_blending_rad - RR_offset †0 AP_blending_rad - AP_offset †0 107) The value of sin(BRang) should be less than 1.0 (Fig. 7.20) (12). sBRang † 1.0 where sBRang=sin(BRang). 108) Angle RR_INang should be positive (Fig. 7.3) (15). -RR_INang †0 (7.63) (7.62) (7.59) (7.60) (7.61) 109) RR_edge should be bigger than dimension distance_from_edge to be able to weld the parts 1 and 2 at the top (Figs. 5.7 and 5.9) (16). RR_h1 + Distance_from_edge*cos(angle_back) - RR_width † 0 (7.64) 275 110) RR_h2 should be smaller than the overall width of roof rail cross section (Fig. 5.9) (17). RR_h2 - RR_width † 0 (7.65) 111,112) To make the angle between the side shell of door and the horizontal direction in the range [-90,90]. Therefore, the absolute value of the cosine of this angle, AP_c, should be less than 1.0 (Fig. 7.5) (24,25). -1.0-AP_c †0 AP_c - 1.0 † 0 (7.66) (7.67) 113,114) A_H, and A_V should be positive so that the A-pillar reinforcement has reasonable shape (Fig. 7.4) (33,34). -A_H †0 -A_V †0 (7.68) (7.69) 115) The projection of A_pillar_blending_rad on global z-axis should be bigger than the horizontal distance measured from the global origin to the edge of flange 1 at the cross section of the header (Fig. 7.21) (35). Projection of A_pillar_blending_rad on z axis – projection of distance between global origin and edge of flange 1 on z † 0 (7.70) 116,117) Both AP_d1, and AP_h2 should be positive so that the shape of the Apillar is reasonable (Fig. 7.4) (36,37). -AP_d1 † 0 - AP_h2 † 0 (7.71) (7.72) 276 • Performance Targets 118-120) The three stiffnesses of the design should be larger than the minimum values. ˆ (K I / O − K I / O ) / K I / O ≤ 0 ˆ (K F / A − K F / A ) / K F / A ≤ 0 ˆ (K − K ) / K ≤ 0 Tor Tor Tor (7.73) (7.74) (7.75) 7.3 Results and Discussion The above optimization problem was solved several times using the nonlinear optimization program DOT (1995). First, we checked if the optimization program converged to a global minimum. Then, we validated the optimization program by comparing the optimization results with FEA results. Finally, results obtained using RSP and NN translators were compared. 7.3.1 Checking the Convergence of the Optimization Program We can use three methods to check whether the optimization program converges to a global minimum. • • • We solve the problem from different initial points to see if the optimization program converges to the same optimum design. We use the final optimum design as an initial point, and solve the problem again to see if the design changes. We also solve the same problem using both the modified feasible direction method (MFD) and the sequential linear programming (SLP) to see if the optimization program converges to the same point. Eight randomly generated designs were used as initial points. The measured stiffness of an actual car joint was used as target. We only considered the mass in the objective function. E.g., α in Eq.7.1 was set to be one. Table 7.2 shows the objective functions of 277 the optimum designs when we start from different initial points. We used SLP to obtain the above results. Both RSP and NN translators were tested. It is observed that the maximum relative difference between the masses of optimum designs is 0.3%. The first 2 to 3 digits of the objective function are the same when starting from different initial designs. In general, the first 2 to 3 digits of design variables of the final designs are found to be the same. Practically, no improvement (less than 0.3%) was achieved by solving the problem again starting from the obtained optimum designs. Finally, it is observed that the optimum design from the NN translator has about 7% larger mass than the designs obtained from the RSP translator. This observation is the same as that for the B-pillar to rocker joint. We also used the modified feasible direction method (MFD) to check the convergence of the optimization program by starting from different initial points. We found that the optimum design obtained using the MFD method was practically the same as that obtained using the SLP method. We chose SLP as the default optimization method in the translator B for the A-pillar to roof rail joint because the SLP method needed fewer iterations than the MFD method. All the optimization results presented in this chapter were obtained using SLP method. The user can easily switch to the MFD method by changing a variable in the optimization program of translator B. 7.3.2 Comparison of the Results of Translator B with FEA Results To validate translator B, we compared the optimization results with FEA results. First, we randomly generated 12 sets of performance targets (Each set contained minimum values for I/O, F/A, and torsion stiffness). Using translator B, we obtained the optimum design corresponding to each set of stiffness requirements. The optimum designs were visually checked using the Pro/Engineer model, and MSC/NASTRAN bulk date files were created. Each model was analyzed and its performance characteristics (stiffnesses and mass) were obtained. The optimization results were compared with the FEA results. Table 7.3a and Figures 7.22a – 7.22d show the results for the I/O, F/A, torsion stiffnesses and the mass when RSP translators were used. The values of design variables of four optimum designs are shown in Table 7.3b. Table 7.3c and Figures 7.23a-7.23d show the results when NN translators were used. Table 7.3d shows the 278 values of design variables of four optimum designs. Table 7.3e shows the correlation coefficients of the 12 designs for the RSP and NN translators. The cross sections of the initial and the optimum designs are compared in Fig. 7.24. By observing the optimum designs, we found the following trends when optimizing the shape of the joint components: A-pillar In general, the size of the section of the A-pillar increases (Fig. 7.24). The shape of the A-pillar section approaches to the shape of a rectangle. The following trends about the values of the design variables are observed (Tables 7.3b and 7.3d) • • • AP_inboard_width reaches its upper bound. This makes the A-pillar section close to a square, and thus increases the stiffness of A-pillar. AP_reinf_inner_door_allowance is close to its upper bound. AP_reinf_flange2_angle assumes a value that makes the outer piece of A-pillar reinforcement almost vertical. Thus, the A-pillar reinforcement approaches the outer shell of the A-pillar. This increases the cross section area and the stiffness of A-pillar. • • Angleback takes its minimum allowable value that is determined by the constraint on spring back angle. AP_blending_rad reaches its lower bound for most designs to reduce the mass of the joint. Header • H_width, H_window_depth (Fig. 5.8) and H_blending_rad (Fig. 5.6) are close to their lower bounds at the optimum. The optimizer tends to reduce the cross section of header to reduce the mass of the joint. This shows that the header branch has higher stiffness compared with the A-pillar branch and it has little effect on the overall stiffness of the joint. 279 Roof rail • RR_width reaches its lower bound. RR_angle_bottom is close to its upper bound. Most dimensions at the roof rail cross section take values between their lower and upper bounds. This shows that the roof rail section is determined by a trade-off between high stiffness and low mass. Global dimensions • Flange1_angle_down (Fig. 5.10) reaches its lower bound. As a result of this trend, the inner plate of part 3 tends to be vertical, which increases the cross section of the A-pillar and the stiffness of the joint (Fig. 7.24). Extrusion/cutting distance • Part3_cut_distance and part4_extension take their minimum allowable values. Since part 3 is thicker than part 4, this will increase the thickness at the bottom of header. • Part7_cut_distance reaches its upper bound. This indicates that part 7 (reinforcement) contributes little to the stiffness of header. The optimizer tries to increase this dimension to reduce the mass of the joint. Thickness • All thicknesses reach their lower bounds. This is reasonable because the optimizer tries to reduce the thickness of each part, and increase the cross section of the A-pillar to achieve the maximum stiffness with the smallest increase of mass. From the comparison of the results of translator B and the FEA results (Figs.7.22a7.23d), we observe that translator B predicts the stiffness and mass accurately when the ˆ ˆ stiffness requirements are lower ( K I / O ≤ 6.5E 7 Nmm, KTor ≤ 1.0 E 7 Nmm , Figs. 7.22aˆ 7.23d). The F/A stiffness requirement, K , does not significantly affect the correlation F/A between the results of translator B and FEA results. Both RSP and NN translators predict F/A stiffness accurately. The RSP translator slightly overestimates the F/A stiffness for most designs. On the other hand, the NN translator slightly underestimates the F/A 280 stiffness for most designs. Both RSP and NN translators B underestimate the mass of the optimum design. E.g., the predicted mass of translator B is lower than the actual mass of the joint. The difference between the predictions of translator B and the FEA results increases when the stiffness requirements are high ˆ ˆ ( K I / O > 6.5 E 7 Nmm , K Tor > 1.0 E 7 Nmm ) . Both the RSP and NN translators B tend to overestimate the stiffness of joint when the stiffness requirements are high. For the RSP translator B, the error of the predicted I/O stiffness can be 24.7% higher than the FEA result (design 6), and error of the predicted torsion stiffness can be 51% higher than the FEA result (design 6). For the NN translator B, the errors of predicted I/O and torsion stiffnesses can be as high as 30.8% (design 12) and 24.3% (design 11), respectively. Both the RSP and NN translators B predict the F/A stiffness and the mass accurately. The errors of the predicted F/A stiffness and the mass are less than 10% and 5%, respectively. The correlation coefficient defined in Eq.6.62 is used to assess the correlation between the FEA and optimization results. Table 7.3e shows the correlation coefficients for the stiffnesses and mass when RSP and NN translators are used. It is found that the correlation coefficients for RSP translators range from 0.8074 to 0.9964. For NN translators, the correlation coefficients range from 0.8919 to 0.9967. The NN translator predicts torsion stiffness more accurately than RSP translator. RSP translator predicts the mass with slightly better accuracy. The predictions of I/O and F/A stiffnesses from RSP and NN translators are equally accurate. Overall, the testing results from NN translators are slightly better than those from RSP translators for the 12 designs considered. By comparing the results obtained using the RSP and NN translators, we observe that the mass of the optimum design obtained using NN translators is typically about 7% higher than that obtained using RSP translators. 7.3.3 Redesign of the Joints of two Cars Using Translator B We used translator B to redesign two actual joints. The measured stiffnesses of the two joints were used as targets. Since the shape of the outer shells of A-pillar to roof rail joint (parts 1 and 2) are generally determined by styling requirements, we fixed the 281 design variables that define the outer shape of A-pillar joint at their values for the actual joint, and optimized only those design variables that define the inner part and reinforcement of the joint. Tables 7.4a and 7.4c show the states and ranges of the design variables of cars A and B. Tables 7.4b and 7.4d present the dimensions of the optimum designs obtained using translator B. Tables 7.4e and 7.4f compare the optimization results with FEA results. We observed that the predictions for F/A stiffness and mass are close to the FEA results if the RSP translators are used. The errors are less than 3% and 2%, respectively. The predictions of torsion stiffness are worse than those for F/A stiffness and mass. The errors are 14.8% and 20.1% for the two cars, respectively. The prediction of I/O stiffness correlates well with FEA results for car A (error is -1.9%), but correlates poorly for car B (error is 16.8%). The NN translators predict the I/O stiffness and mass accurately. The biggest errors are -4.7% and -4.2%, respectively. The errors of predictions of the F/A and torsion stiffnesses for car A are –10.8% and –3.5%, respectively. For car B, they are –6.2% and –11.0%, respectively. Table 7.4f compares the masses of the initial and optimum designs of the two cars. Because we fixed design variables defining the outer shape of the A-pillar joint, the improvement of the optimum design compared with the initial design was small. When RSP translators were used, 7.2% and 11.2% improvement were achieved through optimization for car A and car B. When using NN translators, the improvement for car B was 7.8%. There was practically no improvement for car A. 7.3.4 Parametric Study Figures 7.25 –7.27 show the effects of the I/O, F/A, and torsion stiffness requirements on the mass of the optimum design. To get the above results, we only changed one stiffness requirement at a time and fixed the other two stiffness requirements at the values corresponding to an actual joint. It is observed that, when the required stiffnesses are low, the mass of the optimum design is almost constant or increases very little. The mass begins to increase at a considerable rate with the stiffness after a certain point. We also observe that torsion stiffness requirement has bigger effect on the mass of the optimum designs than the I/O and F/A stiffness requirements. 282 As mentioned in Section 7.2.3, we fixed the lengths of the three branches of A-pillar joint to obtain the previous results. We also studied the effects of the lengths of each branch on the mass of the optimum design. Figures 7.28-7.30 show the effect of lengths of header, roof rail, and A-pillar branches on the mass. It is found that increasing the length of a branch increases the mass of the optimum design. Figures 7.31 and 7.32 show the effects of theta and phi, which define the orientation of the A-pillar, on the mass. It is found that the mass of the optimum designs is insensitive to changes in these angles. We observed in Section 7.3.2 that the thicknesses of several parts reached their lower bounds. Figures 7.33 to 7.36 show the relations between the mass of the optimum design and the lower bounds of thicknesses of parts 1, 2, 3, and 7. We observe that increasing the lower bound of thickness increases the mass of the optimum design. Figure 7.37 shows the effect of increasing the lower bound of RR_width on the mass of the optimum design. As we observed in section 7.3.2, RR_width of the optimum design reached its lower bound. Therefore, increasing its lower bound increases the mass of the optimum design because the region in which the optimizer could search decreases. Door_allowance took a value between its lower and upper bounds at the optimum design. Figures 7.38 and 7.39 show the effects of increasing the lower bound or decreasing the upper bound of door_allowance, respectively, on the mass of the optimum design. As expected, both increase the mass of the optimum design. It is also found that door_allowance contributes little to the mass. Figures 7.40 and 7.41 show the relations between the mass of the optimum design and the lower bounds of H_width and AP_blending_rad. Since both design variables are close to their lower bounds for the optimum design, increasing their lower bounds increases the mass of the optimum design. 283 Finally, it is observed from Figs. 7.25-41 that the NN translator yields heavier designs than the RSP translator for the same stiffness targets. 7.3.5 Discussion of Results Some observations of this study should help designers improve the joint design. For example, this study found that the dimensions of the header branch had little effect on the stiffness of the joint. Therefore, reducing the cross section of header reduces the mass of the joint without reducing the stiffness of the joint. We found that the thickness of every part reached its minimum. As a result of this trend, the mass of the joint decreased. It was also found that most design variables defining the cross section of the roof rail assumed values between their lower and upper bounds. This shows that those design variables were determined by a trade-off between the high stiffness and low mass. In general, the translator B increased the size of the cross section of the A-pillar and reduced the size of the cross sections of other two beams. The reason is that the stiffness of the joint is more sensitive to the properties of the A-pillar than the properties of the other beams. Moreover, the properties of all three beams affect the mass. By comparing the results of the translator B with FEA results, we observed that both the RSP and NN translators B predicted the stiffness and mass accurately when the stiffness requirements were lower. The difference between the predictions of the translators and the FEA results increased as the stiffness requirements increased. Both RSP and NN translators B tended to overestimate the stiffnesses, and underestimate the mass of the optimum design when the stiffness requirements were high. It was found that the NN translator predicted torsion stiffness more accurately than the RSP translator. On the other hand, the RSP translator gave slightly more accurate predictions of the mass. The predictions of the I/O and F/A stiffnesses from RSP and NN translators were equally accurate. The above observations are the same as the conclusion of Chapter 5 for the I/O, F/A stiffnesses and the mass, but different from the conclusion of Chapter 5 for the torsion 284 stiffness. Chapter 5 found that, the RSP translator A was slightly more accurate for the mass. The RSP and NN translators A predicted I/O and F/A stiffnesses equally accurately. The RSP translator A predicted the torsion stiffness more accurately than the NN translator A. However, Chapter 7 found that the prediction of translator B correlated better with the FEA results when the NN translator A was used than when the RSP translator A was used. The conclusion of Chapter 5 was drawn based on the 600 designs in the database for developing translator A. Since we fixed the lengths of the three branches when developing the translator B, only a portion of the 600 designs were in the feasible domain of the translator B. Therefore, the conclusion of this chapter that was drawn from a portion of the 600 designs may be different from the overall conclusion in Chapter 5. It was found that the NN translator yielded heavier designs than the RSP translator. This is because the mass of the optimum design was mainly determined by the torsion stiffness requirement. The difference between the predicted torsion stiffness and the FEA results was bigger when the RSP translator was used than when the NN translator was used. The error of the predicted torsion stiffness from the RSP translator B was much higher than that from the NN translator B when the torsion stiffness requirement was high. According to these results, we recommend to use the NN translator B for the Apillar to roof rail joint. This chapter observed the same trend as that in Chapter 6 for the mass. That is, translator B underestimated the mass of the optimum design. On the other hand, the general trends observed in this chapter are different from those in Chapter 6 for the I/O, F/A and torsion stiffness. In Chapter 6, we found that when the RSP translators were used, the translator B overestimated the stiffness when the stiffness requirements were lower, and underestimated the stiffness when the stiffness requirements were high. When NN translators were used, the translator B tended to overestimate the I/O stiffness when the stiffness requirements were low, but underestimated the I/O stiffness when the stiffness requirements were high. In general, the NN translators B underestimated the F/A and torsion stiffnesses. We found that the RSP translators B gave slightly better predictions than the NN translator B in Chapter 6. The difference of the conclusions of 285 this chapter and Chapter 6 is due to several reasons. First, B-pillar to rocker joint and Apillar to roof rail joint are completely different. A-pillar to roof rail joints vary more from joint to joint than B-pillar to rocker joints. Second, since translator B uses translator A to obtain the responses of a new design in optimization, the accuracy of translator A affects significantly the accuracy of translator B. The optimum design of the B-pillar to rocker joint is mainly determined by the F/A stiffness requirement. Both RSP and NN translators A predict accurately the F/A stiffness of the B-pillar to rocker joint. For the A-pillar to roof rail joint, the predictions of both the RSP and NN translators A are less accurate than those for the B-pillar to rocker joint. The optimizer tends to take advantage of the error in the predictive models (translator A). Therefore, the translator B for the Apillar to roof rail joint is less accurate than that for the B-pillar to rocker joint. Third, when we create the database for developing the translator A of A-pillar joint, we leave the lengths of the three branches be design variables because there is no consistent definition of the length of branches of joint. Since the lengths of the three branches of the A-pillar to roof rail joint affect significantly the stiffnesses and the mass (See Tables 5.55.8), considering the lengths of the three branches as design variables allows the translator A to cover a large range of joint designs. However, this sacrifices the accuracy of the translator A for the A-pillar to roof rail joint. When we develop translator B for Apillar, we fix the lengths of the three branches in order to compare the stiffnesses of different designs fairly. The predictions of translator B would be more accurate if we fixed these three lengths when developing the database and translator A. 286 Table 7.1: Ranges and States of Design Variables Seq. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_ allowance AP_reinf_flange2_angle Ap_blending_rad Bounds (mm,degree) Lower Upper 8.0 8.0 260.00 260.00 105 196 163 240 23.00 23.00 76.00 76.00 15 78 20 20 16 16 16 16 50 50 90 113 90 110 90 130 96 123 90 107 27 63 8 46 78 109 19 34 12 17 96 210 56 146 47 79 39 108 35 107 20 42 53 108 52 78 20 33 25 36 19 35 17 26 0 0 1 9 91 155 22 38 91 55 116 138 State ## Fixed by program ## Dependent ## Dependent ## Dependent # Fixed by user # Fixed by user Independent ## Fixed by program ## Fixed by program ## Fixed by program ## Fixed by program Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent Independent ## Fixed by program Independent Independent Independent Independent Independent 287 40 Part2_cut_distance 55 110 Independent 41 Part3_cut_distance 24 56 Independent 42 Part4_extension 46 90 Independent 43 Part7_cut_distance 44 90 Independent 44 Thickness of Part1 0.81 1.19 Independent 45 Thickness of Part2 1.03 1.14 Independent 46 Thickness of Part3 1.31 1.68 Independent 47 Thickness of Part4 1.00 1.07 Independent 48 Thickness of Part7 1.53 1.83 Independent Lengths of header, roof rail and A-pillar branches are fixed at 260 mm, 213 mm and 161mm, respectively. Table 7.2: Comparison of Optimum Results When Starting From Different Initial Points ( K I / O > 3.16 E 7 Nmm , K F / A > 7.93E 7 Nmm , K Tor > 1.38E 7 Nmm ) Initial Design No. 1 2 3 4 5 6 7 8 Using RSP Translators No. of Function Object Function Evaluations (kg) 574 2.3380 344 2.3433 497 2.3385 500 2.3381 499 2.3382 458 2.3428 575 2.3384 423 2.3383 Using NN Translators No. of Function Objective Function Evaluations (kg) 537 2.4914 804 2.4917 727 2.4914 695 2.4914 614 2.4916 538 2.4915 2403 2.4918 619 2.4920 Table 7.3a: Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint(using RSP Translators) Set No. Stiffness Requirement (Nmm) I/O F/A Tor ×107 ) ×107 ×107 I/O ×107 Optimization Results (Nmm,kg) F/A Torsion ×107 ×107 Mass I/O ×107 FEA Results (Nmm,kg) F/A Torsion ×107 ×107 Mass 1 2 3 4 5 6 7 8 9 10 11 12 3.5 6.5 5.7 6.0 5.0 7.5 4.25 5.5 7.25 5.1 3.4 3.4 5.25 6.75 6.0 8.25 5.0 9.0 4.75 6.5 10. 5.9 8.0 5.9 0.75 1.00 0.8 0.8 0.9 1.8 0.7 0.85 1.4 0.85 1.43 1.1 5.8419 6.8226 5.7024 6.0012 6.0015 9.3395 5.6450 5.8831 7.2816 6.2110 7.1681 7.2490 5.2511 6.7501 6.0018 8.2501 5.0007 9.0000 4.7500 6.5001 10.0020 5.9000 7.9994 5.8996 0.7500 1.0000 0.8003 0.9857 0.9000 1.8001 0.7000 0.8500 1.4006 0.8501 1.4298 1.0998 2.1491 2.2377 2.1877 2.2925 2.1689 2.4828 2.1224 2.2138 2.3897 2.1868 2.3549 2.2312 5.8401 6.8802 5.7973 6.3396 5.8833 7.4874 5.5783 6.1164 7.2926 6.2039 6.7341 6.7114 5.0244 6.3976 5.8579 8.2430 4.9933 8.6355 4.4314 6.4338 9.1500 5.6297 7.7269 5.4363 0.8474 0.9311 0.9137 1.0048 0.8379 1.1917 0.6969 0.9684 1.2760 0.8365 1.0084 0.8483 2.2473 2.3442 2.2910 2.4004 2.2796 2.5443 2.2187 2.3199 2.4785 2.2887 2.4383 2.3261 288 Table 7.3b: Values of Design Variables of Four Optimum Designs From RSP Translator No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_ allowance Design and Value of Design Variable (mm,degree) 1 5 9 12 8.00 8.00 8.00 8.00 260.00 260.00 260.00 260.00 181.65 181.65 190.45 196.00 189.86 189.86 181.76 176.64 23.00 23.00 23.00 23.00 76.00 76.00 76.00 76.00 31.35 31.35 22.55 17.00 20.00 20.00 20.00 20.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 50.00 50.00 50.00 50.00 112.91 92.35 113.00 103.84 90.00 90.00 90.00 90.04 115.00 114.91 114.97 114.97 96.05 96.00 96.06 96.22 106.95 107.00 106.97 106.95 37.94 39.02 57.19 42.89 10.00 10.00 10.00 10.00 78.05 78.01 78.06 78.01 19.00 19.00 19.00 19.00 12.27 12.26 12.22 12.26 96.03 96.02 100.27 96.00 56.00 56.50 56.52 56.49 53.53 54.84 73.23 58.63 43.24 39.00 39.01 39.02 55.99 56.49 56.52 56.49 37.58 39.21 41.98 41.99 78.70 78.66 87.51 93.01 55.35 52.00 71.89 52.01 33.00 32.99 33.00 33.00 26.30 25.01 36.00 25.02 25.50 34.99 35.00 34.97 17.00 21.38 26.00 24.60 0.00 0.00 0.00 0.00 9.00 7.99 7.98 9.00 113.91 114.00 119.42 113.95 34.44 31.74 37.99 33.17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 289 38 39 40 41 42 43 44 45 46 47 48 AP_reinf_flange2_angle Ap_blending_rad Part2_cut_distance Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 116.00 55.01 110.00 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 115.91 55.02 110.00 26.00 46.00 89.91 0.81 1.03 1.31 1.00 1.53 115.97 55.03 110.00 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 115.97 55.02 110.00 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 Table 7.3c: Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint(using NN Translators) No Stiffness Requirement (Nmm) F/A Tor I/O ×107 ×107 ×107 I/O ×107 Optimization Results (Nmm,kg) F/A Torsion ×107 ×107 Mass I/O ×107 FEA Results (Nmm,kg) F/A Torsion ×107 ×107 Mass 1 2 3 4 5 6 7 8 9 10 11 12 3.5 6.5 5.7 6.0 5.0 7.5 4.25 5.5 7.25 5.1 3.4 3.4 5.25 6.75 6.0 8.25 5.0 9.0 4.75 6.5 10. 5.9 8.0 5.9 0.75 1.00 0.8 0.8 0.9 1.8 0.7 0.85 1.4 0.85 1.43 1.1 5.3399 6.4990 5.6990 5.9985 7.2137 9.7608 5.2906 5.4997 7.5898 5.3679 9.1975 8.8012 5.2503 6.7492 6.0001 8.2526 4.9990 9.1894 4.7514 6.4997 10.0020 5.8990 8.3037 5.9000 0.7505 1.0000 0.8070 0.9997 0.9004 1.8005 0.7002 0.8500 1.3998 0.8501 1.4297 1.1000 2.2756 2.3655 2.3134 2.4072 2.3080 2.6286 2.2515 2.3347 2.5211 2.3116 2.5074 2.3892 5.6092 6.7411 6.1530 6.3693 6.4926 8.8787 5.4101 5.8346 7.2727 5.5333 7.3289 6.7297 5.5042 7.0129 6.3124 8.7465 5.4398 9.2253 5.0016 6.9671 10.2960 6.3184 8.3920 6.1323 0.8079 0.9560 0.8459 1.0502 0.8649 1.6295 0.7654 9.1464 1.3939 0.8786 1.1501 0.9872 2.3966 2.4518 2.4194 2.4777 2.4096 2.6582 2.3668 2.4371 2.5965 2.4127 2.5270 2.4481 290 Table 7.3d: Values of Design Variables of Four Optimum Designs From NN Translators No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_ allowance Design and Value of Design Variable (mm,degree) 1 5 9 12 8.00 8.00 8.00 8.00 260.00 260.00 260.00 260.00 181.77 193.72 181.66 196.00 189.75 178.75 189.85 176.65 23.00 23.00 23.00 23.00 76.00 76.00 76.00 76.00 31.23 19.28 31.34 17.00 20.00 20.00 20.00 20.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 16.00 50.00 50.00 50.00 50.00 112.92 112.90 113.00 112.93 90.04 90.02 90.03 90.00 114.95 114.94 114.89 111.88 96.02 96.04 96.02 105.44 90.03 90.09 90.08 90.01 46.59 56.78 52.62 54.17 10.00 10.00 10.00 10.00 78.00 78.00 78.00 78.05 30.00 30.00 30.00 30.00 13.92 13.92 13.92 13.92 96.00 96.00 96.09 96.01 56.03 56.04 56.00 56.04 61.62 71.74 67.62 69.29 46.53 46.53 46.49 41.50 56.03 56.03 55.99 56.04 42.00 42.00 42.00 42.00 108.00 107.99 108.00 94.03 54.09 73.30 62.92 62.61 32.67 33.00 32.93 32.97 32.48 34.08 35.99 36.00 19.00 19.00 19.00 29.07 17.01 25.99 19.85 23.11 0.00 0.00 0.00 0.00 5.05 3.82 2.11 8.99 114.02 130.34 155.00 119.75 37.99 37.96 37.99 38.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 291 38 39 40 41 42 43 44 45 46 47 48 AP_reinf_flange2_angle Ap_blending_rad Part2_cut_distance Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 115.95 55.03 58.91 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 115.94 55.01 109.94 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 115.89 55.03 79.14 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 113.26 124.19 110.00 26.00 46.00 90.00 0.81 1.03 1.31 1.00 1.53 Table 7.3e: Comparison of Correlation Coefficients Obtained using RSP and NN Translators Stiffness/Mass I/O F/A Torsion Mass RSP translators 0.8984 0.9919 0.8074 0.9964 NN translator 0.8919 0.9967 0.9634 0.9841 292 Table 7.4a: State and Ranges of Design Variables of Car A No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_ allowance AP_reinf_flange2_angle State ## Fixed by program ## Dependent ## Dependent ## Dependent # Fixed by user # Fixed by user # Fixed by user ## Fixed by program ## Fixed by program ## Fixed by program ## Fixed by program # Fixed by user Independent # Fixed by user Independent Independent # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user Independent Independent # Fixed by user Independent # Fixed by user Independent # Fixed by user # Fixed by user # Fixed by user Independent Independent Independent Independent Car A Bounds (mm,degree) Lower Upper 8.00 8.00 202.00 202.00 165.00 165.00 163.00 163.00 23.00 23.00 76.00 76.00 15.00 15.00 20.00 20.00 16.00 16.00 16.00 16.00 50.00 50.00 90.00 90.00 90.00 121.00 90.00 90.00 90.00 135.00 90.00 118.00 38.00 38.00 30.00 30.00 80.00 80.00 17.00 37.00 16.00 16.00 95.00 95.00 65.00 65.00 43.00 87.00 40.00 40.00 32.00 118.00 18.00 46.00 90.00 90.00 47.00 86.00 26.00 26.00 23.00 40.00 35.00 35.00 22.00 22.00 0.00 0.00 1.00 29.00 91.00 171.00 20.00 69.00 91.00 144.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 293 39 Ap_blending_rad # Fixed by user 55.00 55.00 40 Part2_cut_distance Independent 50.00 121.00 41 Part3_cut_distance Independent 22.00 62.00 42 Part4_extension Independent 42.00 99.00 43 Part7_cut_distance Independent 40.00 121.00 44 Thickness of Part1 # Fixed by user 0.94 0.94 45 Thickness of Part2 # Fixed by user 1.02 1.02 46 Thickness of Part3 Independent 0.81 1.85 47 Thickness of Part4 Independent 0.81 1.18 48 Thickness of Part7 Independent 1.06 2.01 The lengths of header, roof rail, and A-pillar branches are fixed at 202 mm, 180 mm, and 149.2 mm, respectively. Table 7.4b: Optimum Design for Car A No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom Optimum Design for Car 1 (mm,degree) Use RSP Use NN 8.00 8.00 202.00 202.00 165.00 165.00 163.01 163.01 23.00 23.00 76.00 76.00 15.00 15.00 20.00 20.00 16.00 16.00 16.00 16.00 50.00 50.00 90.00 90.00 90.00 90.00 90.00 90.00 135.00 90.07 117.97 92.91 38.00 38.00 30.00 30.00 80.00 80.00 21.00 30.00 16.00 16.00 95.00 95.00 65.00 65.00 55.73 67.36 40.00 40.00 56.92 61.94 42.03 45.20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 294 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_all owance AP_reinf_flange2_angle Ap_blending_rad Part2_cut_distance Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 90.00 59.73 26.00 32.30 35.00 22.00 0.00 1.00 170.01 37.03 91.00 55.00 121.00 22.01 42.01 121.00 0.94 1.02 1.73 0.91 1.13 90.00 61.98 26.00 36.71 35.00 22.00 0.00 1.00 168.57 31.47 91.00 55.00 50.00 22.00 42.00 121.00 0.94 1.02 1.68 0.91 1.17 295 Table 7.4c: States and Ranges of Design Variables of Car B No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_all owance AP_reinf_flange2_angle Ap_blending_rad State ## Fixed by program ## Dependent ## Dependent ## Dependent # Fixed by user # Fixed by user # Fixed by user ## Fixed by program ## Fixed by program ## Fixed by program ## Fixed by program # Fixed by user Independent # Fixed by user Independent Independent # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user # Fixed by user # Fixed by user Independent # Fixed by user Independent Independent # Fixed by user Independent # Fixed by user Independent # Fixed by user # Fixed by user # Fixed by user Independent Independent Independent Independent # Fixed by user Car B Bounds (mm,degree) Lower Upper 8.00 8.00 255.00 255.00 196.00 196.00 179.00 179.00 25.00 25.00 76.00 76.00 37.00 37.00 20.00 20.00 16.00 16.00 16.00 16.00 50.00 50.00 103.00 103.00 90.00 121.00 130.00 130.00 90.00 135.00 90.00 118.00 62.00 62.00 23.00 23.00 90.00 90.00 17.00 37.00 15.00 15.00 140.00 140.00 73.00 73.00 43.00 87.00 52.00 52.00 32.00 118.00 18.00 46.00 88.00 88.00 47.00 86.00 31.00 31.00 23.00 40.00 19.00 19.00 26.00 26.00 10.00 10.00 1.00 29.00 91.00 171.00 20.00 69.00 91.00 100.00 144.00 100.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 296 40 41 42 43 44 45 46 47 48 Part2_cut_distance Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 Independent Independent Independent Independent # Fixed by user # Fixed by user Independent Independent Independent 50.00 22.00 42.00 40.00 0.89 0.89 1.12 0.81 1.06 121.00 62.00 99.00 121.00 0.89 0.89 1.12 1.18 2.01 The lengths of header, roof rail and A-pillar branches are fixed at 255 mm, 233 mm, and 145.5 mm, respectively. Table 7.4d: Optimum Design for Car B No Name of Design Variables Distance_from_edge Header_offset Roof_rail_offset A_pillar_offset Theta Phi Header_horizontal_offset Flange1_width Flange2_width Flange3_width Spot_weld_spacing Flange1_angle_up Flange1_angle_down Flange2_angle_up Flange2_angle_down Flange3_angle_down Door_allowance Angleback H_width H_height H_window_depth H_blending_rad RR_width RR_height RR_h1 RR_h2 RR_angle_bottom Optimum Design for Car 2 (mm,degree) Use RSP Use NN 8.00 8.00 255.00 255.00 196.00 196.00 179.03 179.03 25.00 25.00 76.00 76.00 37.00 37.00 20.00 20.00 16.00 16.00 16.00 16.00 50.00 50.00 103.00 103.00 98.65 90.89 130.00 130.00 135.00 134.98 118.00 105.94 62.00 62.00 23.00 23.00 90.00 90.00 19.94 30.00 15.00 15.00 140.00 140.00 73.00 73.00 67.14 73.01 52.00 52.00 73.00 73.00 18.00 25.08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 297 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 RR_blending_rad AP_height AP_inboard_width AP_inboard_depth AP_door_ws_distance AP_window_depth AP_flange1_angle AP_reinf_depth AP_reinf_flange1_angle AP_reinf_inner_door_all owance AP_reinf_flange2_angle Ap_blending_rad Part2_cut_distance Part3_cut_distance Part4_extension Part7_cut_distance Thickness of Part1 Thickness of Part2 Thickness of Part3 Thickness of Part4 Thickness of Part7 88.00 70.57 31.00 39.90 19.00 26.00 10.00 1.00 159.25 37.26 144.00 100.00 121.00 22.00 42.00 121.00 0.89 0.89 1.12 0.91 1.06 88.00 70.57 31.00 38.12 19.00 26.00 10.00 1.00 158.46 35.30 131.00 100.00 83.28 22.00 42.00 121.00 0.89 0.89 1.12 0.91 1.06 Table 7.4e: Comparison of Optimization and FEA Results for A-pillar to Roof Rail Joint using RSP and NN Translators Cars/ Stiff Req. Car A I/O>6.531E7 F/A>4.929E7 Tor>1.259E7 Car B I/O>3.399E7 F/A>5.981E7 Tor>1.044E7 Stiffness / Mass I/O F/A Torsion Mass I/O F/A Torsion Mass Use RSP Translators (Nmm,kg) FEA Predictions (err) 6.6568E7 6.5329E7 (-1.9%) 6.5412E7 6.3504E7 (-2.9%) 1.1171E7 1.2826E7 (14.8%) 1.8835 1.8478 (-1.9%) 4.4762E7 5.2274E7 (16.8%) 5.8044E7 5.9812E7 (3.0%) 0.8766E7 1.0524E7 (20.1%) 2.1667 2.1528 (-0.6%) Use NN Translators (Nmm,kg) FEA Predictions (err) 6.6554E7 6.5301E7 (-1.9%) 7.7032E7 6.8737E7 (-10.8%) 1.3047E7 1.2587E7 (-3.5%) 2.0479 2.0279 (-1.0%) 5.4829E7 5.2274E7 (-4.7%) 6.3793E7 5.9812E7 (-6.2%) 1.1822E7 1.0524E7 (-11.0%) 2.2477 2.1528 (-4.2%) 298 Table 7.4f: Comparison of Mass of the Initial and Optimum Designs for Car A and Car B Cars Initial Design (kg) 2.0302 2.4396 Optimum Design (kg) and Percentage Improvement RSP NN 1.8835 (7.2%) 2.0479 (-0.9%) 2.1667 (11.2%) 2.2477 (7.8%) Car A Car B 299 Length of roof rail Length of A-pillar Up Length of header (Header_offset) Front O Outboard Roof_rail_offset A_pillar_offset Figure 7.1: Definition of Lengths of Three Branches for A-pillar to Roof Rail Joint 300 Width of RR cross section Minimum angel to avoid spring back Figure 7.8: Explanations of Spring Back Angle 307 H_L1 H_L2 H_L5 H_L3 H_L4 Up Front Figure 7.9: Manufacturing Constraints on Lengths of Cross Section of Header 308 H_A1 H_A3 H_A2 Up Front Figure 7.10: Manufacturing Constraints on Angles of Cross Section of Header 309 RR_L1 RR_L2 RR_L6 RR_L3 Up RR_L5 Outboard RR_L4 Figure 7.11: Manufacturing Constraints on Lengths of Cross Section of Roof Rail 310 RR_A1 RR_A2 RR_A7 RR_A6 RR_A3 Up RR_A4 Outboard RR_A5 Figure 7.12: Manufacturing Constraints on Angles of Cross Section of Roof Rail 311 B RR_Sp1 RR_Sp3 B Up A Outboard A RR_Sp2 A’ A’ B’ B’ AA // A’A’ BB // B’B’ RR_AD1 Figure 7.13: Constraints on the Spring Back Angles and Die Lock Angle of Roof Rail Cross Section 312 AP_L2 AP_L1 AP_L8 AP_L7 AP_L6 AP_L9 AP_L3 Outboard AP_L5 AP_L4 Figure 7.14: Manufacturing Constraints on Lengths of A-pillar Section 313 AP_A2 AP_A1 AP_A3 AP_A8 AP_A7 Outboard AP_A6 AP_A4 AP_A5 Figure 7.15: Manufacturing Constraints on Angles of the Outer Shell of A-pillar Cross Section 314 AP_A9 AP_A10 AP_A11 Outboard AP_A12 Figure 7.16: Manufacturing Constraints on Angles of A-pillar Reinforcement 315 AP_Sp1 A A AP_Sp6 AP_Sp5 A’ A’ B’ AP_Sp4 AP_AD3 AA // A’A’ BB // B’B’ CB // C’C’ DA // D’D’ B’ AP_Sp7 AP_Sp2 B D B AP_AD2 AP_Sp3 D’ C’ C’ C Outboard D’ Figure 7.17: Constraints on Spring Back Angles and Die Lock Angles of A-pillar Cross Section 316 RR_edge AP_edge Figure 7.18: Styling Constraint 317 Header Top Front Blind Angle Driver’s eye Figure 7.19: Explanation of Safety Constraint 318 Up Outboard BRang Figure 7.20: Constraints on an Angle of Roof Rail Cross Section 319 Projection of distance between global origin and edge of flange 1 on axis z #Projection of AP_blending_rad at axis z z O x AP_blending_rad Figure 7.21: Constraint on A-pillar Blending Radius 320 1.0E+08 6 9.0E+07 Optimization Results 8.0E+07 12 9 7.0E+07 5 10 4 8 11 2 6.0E+07 7 1 3 Using polynomial translators Correlation coefficient = 0.8984 5.0E+07 4.0E+07 4.0E+07 5.0E+07 6.0E+07 7.0E+07 FEA Results 8.0E+07 9.0E+07 1.0E+08 Figure 7.22a: Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the I/O Stiffness of A-pillar to Roof Rail Joint 321 1.2E+08 1.1E+08 9 Optimization Results 9.0E+07 4 11 6 7.5E+07 2 8 12 10 3 1 7 5 6.0E+07 Using polynomial translators Correlation coefficient = 0.9919 4.5E+07 3.0E+07 3.0E+07 4.5E+07 6.0E+07 7.5E+07 9.0E+07 1.1E+08 1.2E+08 FEA Results Figure 7.22b: Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the F/A Stiffness of A-pillar to Roof Rail Joint 322 2 .0 E + 0 7 6 1 .6 E + 0 7 O p t im iz at io n R es u lt s 11 9 1 .2 E + 0 7 12 2 5 10 8 3 4 Using polynomial translators Correlation coefficient = 0.8074 8 .0 E + 0 6 7 1 4 .0 E + 0 6 4 .0 E + 0 6 8 .0 E + 0 6 1 .2 E + 0 7 1 .6 E + 0 7 2 .0 E + 0 7 F E A R es u lt s Figure 7.22c: Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the Torsion Stiffness of A-pillar to Roof Rail Joint 323 3 2.9 2.8 2.7 2.6 2.5 O p timizatio n Results 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 F EA Results 7 4 3 1 5 10 12 8 2 11 9 6 Using polynomial translators Correlation coefficient = 0.9964 Figure 7.22d: Comparison of FEA Results and Optimization Results Obtained using RSP Translator for the Mass of A-pillar to Roof Rail Joint 324 1 .1 E +0 8 1 .0 E +0 8 11 6 9 .0 E +0 7 12 O ptimization Results 8 .0 E +0 7 9 5 7 .0 E +0 7 2 4 6 .0 E +0 7 1 8 10 3 U s in g n e u ra l n e tw o rk tra n s la to rs Co rre la tio n c o e ffic ie n t = 0.8919 5 .0 E +0 7 7 4 .0 E +0 7 4 .0 E +0 7 5 .0 E +0 7 6 .0 E +0 7 7 .0 E +0 7 8 .0 E +0 7 9 .0 E +0 7 1 .0 E +0 8 F EA Results Figure 7.23a: Comparison of FEA Results and Optimization Results Obtained using NN Translator for the I/O Stiffness of A-pillar to Roof Rail Joint 325 1 .2 E +0 8 1 .1 E +0 8 6 9 O ptimization Results 9 .0 E +0 7 11 4 7 .5 E +0 7 2 8 10 3 12 1 7 5 6 .0 E +0 7 U s in g n e u ra l n e tw o rk tra n s la to rs Co rre la tio n c o e ffic ie n t = 0.9967 4 .5 E +0 7 3 .0 E +0 7 3 .0 E +0 7 4 .5 E +0 7 6 .0 E +0 7 7 .5 E +0 7 9 .0 E +0 7 1 .1 E +0 8 1 .2 E +0 8 F EA Results Figure 7.23b: Comparison of FEA Results and Optimization Results Obtained using NN Translator for the F/A Stiffness of A-pillar to Roof Rail Joint 326 2 .0 E +0 7 6 1 .6 E +0 7 O ptimization Results 11 9 1 .2 E +0 7 12 2 5 4 8 .0 E +0 6 3 7 1 8 10 U s in g n e u ra l n e tw o rk tra n s la to rs Co rre la tio n c o e ffic ie n t = 0.9634 4 .0 E +0 6 4 .0 E +0 6 8 .0 E +0 6 1 .2 E +0 7 1 .6 E +0 7 2 .0 E +0 7 F EA Results Figure 7.23c: Comparison of FEA Results and Optimization Results Obtained using NN Translator for the Torsion Stiffness of A-pillar to Roof Rail Joint 327 3 2 .9 2 .8 2 .7 2 .6 11 6 O ptimization Results 2 .5 2 .4 2 .3 2 .2 2 .1 2 1 .9 1 .8 1 .7 1 .6 1 .5 1 .5 1 .6 1 .7 1 .8 1 .9 2 2 .1 2 .2 2 .3 2 .4 2 .5 2 .6 2 .7 2 .8 2 .9 3 7 12 4 13 8 2 10 u s e n e u ra l n e tw o rk tra n s la to rs Co rre la tio n c o e ffic ie n t = 0.9841 F EA Results Figure 7.23d: Comparison of FEA Results and Optimization Results Obtained using NN Translator for the Mass of A-pillar to Roof Rail Joint 328 Optimum Design (solid line) Initial Design (dashed line) Figure 7.24: Comparison of A-pillar Cross Sections of Actual Joint and Optimum Joint from Translator B 329 3 2.9 2.8 2.7 2.6 2.5 2.4 M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 2.0E+ 07 3.0E+ 07 4.0E+ 07 5.0E+ 07 6.0E+ 07 7.0E+ 07 8.0E+ 07 9.0E+ 07 1.0E+ 08 P o ly n o mia l R e s u lts N N R e s u lts I/O S tiffn e s s R e q u ire m e n t Figure 7.25: Mass of Optimum Design vs. I/O Stiffness Requirement (KF/A –7.3234E7, KTor – 1.0945E7) 330 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 2.0E+07 3.0E+07 4.0E+07 5.0E+07 6.0E+07 7.0E+07 8.0E+07 9.0E+07 1.0E+08 F /A S tiffn e s s R e q u ire me n t Figure 7.26: Mass of Optimum Design vs. F/A Stiffness Requirement (KI/O – 5.3936E7, KTor – 1.0945E7) 331 3.5 3.4 3.3 3.2 3.1 3 2.9 2.8 2.7 2.6 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 4.0E+06 8.0E+06 1.2E+07 1.6E+07 2.0E+07 2.4E+07 T o rs io n S tiffn e s s R e q u ire me n t Figure 7.27: Mass of Optimum Design vs. Torsion Stiffness Requirement (KI/O – 5.3936E7, KF/A – 7.3234E7) 332 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 160 180 200 220 240 260 280 300 H _ o ffs e t Figure 7.28: Mass of Optimum Design vs.. H_offset (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 333 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 150 170 190 210 230 250 270 Le n g th o f R o o f R a il B ra n c h Figure 7.29: Mass of Optimum Design vs. Length of Roof Rail Branch (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 334 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 130 140 150 160 170 180 190 200 210 220 Le n g th o f A -p illa r B ra n c h Figure 7.30: Mass of Optimum Design vs. Length of A-pillar Branch (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 335 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 20 21 22 23 24 25 26 27 28 29 30 T h e ta Figure 7.31: Mass of Optimum Design vs. Angle Theta (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 336 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 71 72 73 74 75 76 77 78 79 80 81 Ph i Figure 7.32: Mass of Optimum Design vs. Angle Phi (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 337 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Lo w e r B o u n d o f T h ic kn e s s o f P a rt 1 Figure 7.33: Mass of Optimum Design vs. Thickness of Part 1 (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 338 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.85 0.9 0.95 1 1.05 1.1 1.15 Lo w e r B o u n d o f T h ic kn e s s o f P a rt 2 Figure 7.34: Mass of Optimum Design vs. Thickness of Part 2 (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 339 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.89 0.99 1.09 1.19 1.29 1.39 1.49 1.59 1.69 Lo w e r B o u n d o f T h ic kn e s s o f P a rt 3 Figure 7.35: Mass of Optimum Design vs. Thickness of Part 3 (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 340 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Lo w e r B o u n d o f T h ic kn e s s o f P a rt 7 Figure 7.36: Mass of Optimum Design vs. Thickness of Part 7 (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 341 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 50 60 70 80 90 100 110 120 130 140 150 Lo w e r B o u n d o f R R _ w id th Figure 7.37: Mass of Optimum Design vs. the Lower Bound of RR_width (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 342 3 2.9 2.8 2.7 2.6 2.5 2.4 P o ly n o mia l R e s u lts N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 25 30 35 40 45 50 55 60 65 Lo w e r B o u n d o f D o o r_ a llo w a n c e Figure 7.38: Mass of Optimum Design vs. the Lower Bound of Door_allowance (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 343 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 25 30 35 40 45 50 55 60 65 U p p e r B o u n d o f D o o r_ a llo w a n c e Figure 7.39: Mass of Optimum Design vs. the Upper Bound of Door_allowance (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 344 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 75 80 85 90 95 100 105 110 Lo w e r B o u n d o f H _ w id th Figure 7.40: Mass of Optimum Design vs. the Lower Bound of H_width (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 345 3 2.9 2.8 P o ly n o mia l R e s u lts 2.7 2.6 2.5 2.4 N N R e s u lts M as s 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 55 65 75 85 95 105 115 125 135 Lo w e r B o u n d o f A P _ b le n d in g _ ra d Figure 7.41: Mass of Optimum Design vs. the Lower Bound of AP_blending_rad (KI/O – 5.3936E7, KF/A – 7.3234E7, KTor – 1.0945E7) 346 Chapter 8 Conclusions 8.1 Conclusions This study focused on the development of two translators for the design guidance of joints. General methodologies for developing translator A and translator B for all car joints were presented. The methodologies were demonstrated on the B-pillar to rocker and A-pillar to roof rail joints. Translator A can predict the stiffness and mass of a given joint almost instantaneously with good accuracy. Translator B designs the most efficient feasible joint that satisfies given performance (stiffness) targets. The parametric model of the B-pillar to rocker joint created by Murphy (1995) was improved to make it more robust. A parametric model for the A-pillar to roof rail joint was created using Pro/Engineer. Using the B-pillar to rocker and the A-pillar to roof rail models, the stiffness and mass of a few actual joints were estimated and compared with experimental results. For the B-pillar joint, the error in the predictions of the I/O, F/A, and torsion stiffnesses were within the range in which measurements can vary due to hardware variability and experimental errors. For the A-pillar to roof rail joint, the Pro/Engineer model approximated the I/O and torsion stiffness well. The estimated values of F/A stiffness from the FEA model did not agree well with experimental results (The error was about 40% higher than the measured stiffness for a joint). This is due to several reasons. First, since there is no widely accepted definition of length of a branch, the three branches of a joint may be cut at different lengths than those used in the analysis. Second, since the orientation of I/O and F/A stiffnesses are difficult to measure exactly, test engineers often assume them to be in the horizontal and vertical directions. As a result, stiffnesses are defined differently in FEA and tests. Finally, the model used in FEA includes many simplifications. 347 A database was created for each joint by modifying the design variables of the joint randomly and by calculating the stiffnesses and masses of the resulting designs. Methods for design of experiments (DOE) are generally more efficient than a random method for creating designs. However, they cannot be directly applied to problems that involve many constraints that restrict the region of the design variables. We studied the ranking of design variables in terms of importance on the stiffness and mass of the joint using the database and stepwise regression. F-ratio, which measures the relative importance of design variables, was obtained using a linear polynomial model. The design variables with the biggest F-ratio affect significantly the stiffness or mass. The important design variables were identified and ranked according to the value of Fratio. These results can help improve design of joints. Translator A was created using both response surface polynomials and neural networks. Different polynomial models, including, linear polynomial models, second degree polynomial models, and double regression models, were studied. A second regression is performed in the double regression using the results from the linear polynomial model to improve the prediction. Three layer neural networks, which have one input layer, one hidden layer and one output layer, were used to create the neural network translators. Cross-validation was used in this study to avoid overtraining the neural networks. Specifically, all designs were split into three groups. The first group was used to train a neural network. The second group was used to determine when to stop training the neural network. The training process stopped when the error corresponding to the second group reaches its minimum. The third group was used to test the generalization performance of the trained neural network. Different neural network architectures (the number of input design variables, and the number of neurons in the hidden layer) were studied, from which the one with the best generalization performance was chosen to be the neural network translator. Translator A can greatly reduce the time of analyzing a joint design and thus enables designers to optimize a joint. Two optimization programs (translators B) were developed. These tools find the best, feasible design when the user specifies the stiffness requirements and the ranges of the 348 design variables. Translator A was incorporated into the program of translator B. The user can use the polynomial translator A or the neural network translator A. Translator B accounts for manufacturing, packaging, styling, performance and mathematical constraints. These constraints help ensure that the optimum design from translator B is feasible and its stiffness and mass are predicted with reasonable accuracy by the translator A. Two optimization algorithms, namely, the modified feasible direction method (MFD) and sequential linear programming (SLP), were tested in translator B. SLP is the default algorithm since it generally converges faster than the MFD in the problems considered. The user can easily switch optimization methods in the program. Several methods were used to validate translator B. First, optimum designs from translator B were checked to make sure that they corresponded to a global minimum by starting from different initial points. In all cases, the translator B yielded the same design. Both MFD, SLP and another optimization program written in Mathematica were used to solve the same problem. The solutions were almost identical in all cases considered. Second, a number of random stiffness requirements were generated. The optimum designs corresponding to each set of stiffness requirements were obtained using translator B. Then, the optimum results were compared with FEA results The two translators developed in this study should help set up meaningful performance targets for joints. They should also improve designs of new car models because they allow designers analyze many designs rapidly. This study shows that: a) We can simulate FEA for predicting the stiffness of joints using a polynomial or a neural network. b) It is easier to predict the stiffness and mass of the B-pillar to rocker joint than the stiffness and mass of the A-pillar to roof rail joint. c) It is much easier to predict the mass than the stiffness. d) The neural network fitted the data used for training slightly better than the polynomial. However, the accuracy of polynomial deteriorated less than that of the neural network when both models were used to analyze designs they had not seen before. 349 f) Based on the results, we cannot conclude that either the response surface polynomials or the neural networks were consistently more accurate in predicting the stiffness and mass of designs that they had not seen before. The above conclusions are different from these reached by Zhu (1994). The reason is that Zhu used complete second degree polynomials, which contained redundant terms. This study used stepwise regression, which filtered out these terms. g) In general, when the stiffness target are too high, translator B yields designs whose true stiffness can be significantly different from the predicted and the target stiffnesses. However, there is not such problem when the stiffness targets are reasonable. h) The neural network translator B consistently yielded heavier designs than the polynomial translator B. i) Translator B was found to be efficient and robust. The reason is that it uses polynomial or neural network models to analyze a design, which are much more efficient than FEA 8.2 Recommendations for Future Work Following are a few tasks that can improve the methodologies for the translators and facilitate their integration to the design process in the automotive industry. a) Develop consistent definitions of some joint parameters There is no widely acceptable definition of many parameters of joints. In many cases, these parameters affect seriously stiffness calculations and test results. This lack of consistency can be a serious problem when comparing joints and setting stiffness targets. It is important to establish a protocol for analysis and testing of joints. Specifically, we need establish consistent definitions of the following: • • • The joint center The length(s) of the joint branches Reference point(s) relative to which the lengths of the branches must be measured 350 • • The orientations of cutting planes The orientations of the planes about which the branches are assumed to rotate when calculating stiffness The parameters used to define the above quantities should be meaningful to designers and easy to measure. b) Automating the process for developing a database for translator A Developing a database of examples that are used to develop translator A was very time consuming. It took two to three months to develop such a database. The reason is mainly that many designs created by randomly changing the design variables are found infeasible. Moreover, some designs cannot be simulated by the CAD software although they are feasible. Since in industries, the designer may need to design and/or improve a joint in a much shorter time, it will be helpful to automate the process of developing the database. The automation of such process requires to interface different software packages. Because of the limitations of CAD software, it is still a difficult task to develop an error free program that can automate such work. Some progress has been achieved that can get the FEA information for more than 50 designs in a few hours. Once the automating program is finished, it will reduce the time of developing a database to a few days. c) Universal translators The methodology developed in this dissertation is for constructing a translator that works only for a specific joint architecture. Thus, an engineer has to develop a new translator when he/she wants to consider a new architecture. A universal translator, which can process many types of joint architectures, should be more effective and economical for design of joints. This task should establish a methodology for a universal translator that can predict the performance characteristics of most types of joint architectures that are important to the automotive industry. The translator should be fast (analyze a joint at a small fraction of a 351 second), and accurate (average error in stiffness and mass should be less than 10% to 15%). 8.3 Deliverables The deliverables of the study presented in this dissertation include the methodology for developing translators, the parametric models of two joints and the translators A and B for these joints. Our B-pillar to rocker model and translators A and B for the B-pillar to rocker joint are used by an automotive company. 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Zhu, M. “Translators for Design Guidance of Joints in Automotive Structures,” Ph.D. Dissertation, Department of Aerospace and Ocean Engineering , Virginia Polytechnic Institute and State University, 1994. 360 Vita Luohui Long was born in Luoyang, Henan province, China on January 21, 1969. In 1990 he graduated from National University of Defense Technology with a Bachelor of Science degree in Applied Mechanics. Since 1990, he studied at Beijing Institute of Astronautical Systems Engineering as a graduate student. He earned his Master of Science degree in Structural Design and Structural Optimization from Beijing Institute of Astronautical Systems Engineering. After graduation in 1993, he worked in the same institute as a research engineer for one and a half years. In August 1994 he continued his education at Virginia Polytechnic Institute and State University in Blacksburg, Virginia. While studying at Virginia Polytechnic Institute and State University he worked as a cooperative student at American Bureau of Shipping for six months. In 1998 he graduated from Virginia Polytechnic Institute and State University with a Doctor of Philosophy degree in Aerospace Engineering. 361